How does Zumberge's 1981 gravitational measurements relate to gravitational waves?
This represents a major misunderstanding of what a gravitational wave is. The effect presented is simply the semi-static gravitational field at earth due to the earth, moon and sun. It is predicted by Newtonian gravity. There is no 'wave' that propagated, it's the instant positions of the 3 bodies that change over 1 day (and over 1 year also).
It does not show that the change moved at the speed of light, which gravitational waves do. Nothing in Newton's equations talk about the speed of light. The GR equations for 3 bodies moving like the earth-sun-moon can only be solved approximately, and in this case it'd be through a post-Newtonian approximation. The pseudo-static term(s) would be the same but possibly some GR correction - and if it is (And I'm not sure if the strongest term correction might not be something like the term for the perihelion of mercury, or something else, in any case extremely small and not measurable in their g measurement). But that's not even a grav wave. The grav waves would be even smaller probably - you'd have to compute the rate of change of the quadrupole moment of the configuration, and do some other calculations. The simpler problem of just the grav radiation of the earth-sun rotation around each other gives a resultant power dissipated that translates in the orbit of the earth loosing altitude ('altitude' above the sun) of the size of 1 proton per day. That g change they measured in your graph is about 10 to the minus 7 g's. It isn't even dissipative, as the bodies keep doing the same thing over and over, in your approximation. If you don't see that dissipation you are not seeing the gravitational waves.
There is probably many other ways to see that what you're discussing, what the graphic measured, is not a gravitational wave, but rather a very slow change in a static gravity field, the one produced by the 3 bodies.
Grav waves produce something different than just a change in gravity in one direction, they do it in 2 directions at once, an asymmetrical squeezing of a circle first in one axis and then in the other, like squeezing a balloon in one direction, making it bulge in the other.
Like Nathaniel said, it's like comparing a (semi) static electric field (say produced by rubbing a couple rags together) and moving them around some, with light.
Note: yes, even changing static fields can not produce a change in what's observed at a distance faster than the speed of light, but that doesn't come in at all in your graphic, too small a differential effect for it to see it.
[Note: I work on gravitational waves, and am an author on several of the recent LIGO papers on GW150914 — though I am not a member of the LIGO collaboration. So if you're looking for conspiracy theories, that can be your reason to ignore me.]
Zumberge, Rinker, and Faller (ZRF) did not measure gravitational waves. To explain this, I'll start off with an analogy, then discuss gravitational waves directly, followed by the actual math from Einstein's equations that shows very explicitly what I'm talking about.
But first, I'll briefly point out the most obvious and (to me) persuasive point: ZRF found variations at about twice the frequency of the Earth's rotation — consistent with the usual tides, rather than gravitational waves. But Einstein's theory tells us that gravitational waves will only be produced strongly by the orbit of a pair of objects. The moon is orbiting earth on a much slower timescale, so the frequency of gravitational waves from its orbit will be roughly 27 times lower. The earth is orbiting on an even slower timescale, so the frequency of gravitational waves from its orbit will be roughly 365 times lower. Thus, ZRF's data are not measuring the dynamics spacetime effects that Einstein called gravitational waves.
Analogy
The first point to understand is that variations are not the same things as waves. As I drive across hilly roads, my car's altitude varies up and down, but the hills are stationary; I haven't discovered seismic waves. You can argue that seismic waves would also show up in the data, just at a much smaller level and with different frequencies. That's true in principle, but my data are not precise enough to measure those quantities. My results are consistent with both the presence and absence of seismic waves. And more importantly, the point is that seismic waves are motion of the ground itself, whereas the plot of my car's altitude just shows how I moved across terrain that — for all I know — was perfectly stationary.
Gravitational waves
In exactly the same way, gravitational waves are disturbances of spacetime itself, rather than variations I measure as I move across a (nearly) static spacetime. The variations ZRF saw were consistent with a nearly static gravitational field — no waves. In particular, what they saw was consistent with both Newtonian gravity and Einstein's General Relativity, while gravitational waves are only consistent with General Relativity. Rather than the gravitational field changing in time at a particular place (which is what gravitational waves cause), ZRF's measuring device was moving through that field, because the device was moving along with the earth's surface through the field. This is why they saw variations in time, but those variations were not gravitational waves. [More technically, the OP seems to be confusing an advective derivative and a partial derivative.] The ZRF measurement looks like a very nice result, but it's just a different phenomenon.
To be more specific, gravitational waves are not just disturbances in spacetime itself; they are disturbances that propagate as waves at the speed of light. We know that the GW150914 signal moved — at least nearly — at the speed of light, because it was measured in one detector 7 milliseconds after the other. And since the detectors are located 10 milli-light-seconds apart, that means the signal traveled at least nearly as fast as light. [We believe that the signal traveled at the speed of light; we attribute the difference to the plane nature of these gravitational waves and the fact that the waves weren't going straight from one detector to the other. See here for an explanation.]
Math from Einstein's Equations
So now let's look at the key mathematics underpinning this definition. This is discussed in a section on wikipedia (which I think I wrote, actually). The basic point is that the gravitational field is described by the quantity $\bar{h}^{\alpha\beta}$, which obeys the equation \begin{equation} \tag{1} \frac{1}{c^2} \frac{\partial^2} {\partial t^2} \bar{h}^{\alpha\beta} = \nabla^2 \bar{h}^{\alpha\beta}. \end{equation} (At least this is true in empty space, and ignoring nonlinearities.) If you've taken a college-level physics course, you should recognize this equation as a basic wave equation. The left-hand side is measuring the rate of change in time at a particular place, while the right-hand side is measuring the rate of change as you move around in space at a particular time. As seen on that wiki page, this equation is derived directly from Einstein's equations. In particular, Einstein predicted equation (1), with $c$ as the speed of light. It turns out that Newton's theory of gravity assumes that the field propagates infinitely fast: $c \to \infty$, which is equivalent to $1/c^2 \to 0$. This means that in Newtonian gravity, you can ignore the left-hand side, which is the rate of change of the field at a particular place with respect to time. And indeed it turns out that the gravity of the solar system is pretty well described by that simpler Newtonian equation \begin{equation} \tag{2} \nabla^2 \bar{h}^{\alpha\beta} = 0. \end{equation} In fact, this is essentially the same as the well known Poisson's equation as it appears in Newtonian gravity. Now, $\bar{h}^{\alpha \beta}$ is still allowed to change in time, because the planets and moons (and sun) can move, but the time-dependence of $\bar{h}^{\alpha \beta}$ simply doesn't come into this equation.
So we can look at the field that ZRF were measuring. Their data are well modeled [as Floris showed beautifully here] by a fairly simple field that is pretty well specified as \begin{equation} \tag{3} \bar{h}^{00} = 2 \frac{G_\mathrm{N}} {c^2} \left( \frac{M_\mathrm{Earth}} {d_\mathrm{Earth}} + \frac{M_\mathrm{Moon}} {d_\mathrm{Moon}} + \frac{M_\mathrm{Sun}} {d_\mathrm{Sun}} \right), \end{equation} where $d_\mathrm{Earth}$ is the distance from the center of the Earth to the specified point, etc. [Floris's model is more sophisticated than this, with its Love numbers and whatnot, but this is enough to get the point across.] Now, this formula already satisfies equation (2) — Newton's theory — but it also very nearly satisfies equation (1), because the only time dependence involves how quickly the earth, moon, or sun are accelerating towards or away from the point in question — which are all very small numbers. Then, that's divided by $c^2$, which is a huge number, so the left-hand side of equation (1) is basically zero anyway. Plugging in some numbers I get at most \begin{equation} \frac{1}{c^2} \frac{\partial^2} {\partial t^2} \bar{h}^{00} \approx 10^{-30} \mathrm{m}^{-2}. \end{equation} So to satisfy Einstein's equation (1), you just have to add something to equation (3) to balance this out, which would be a really tiny quantity — well within the apparent errors in ZRF's data. So here's the key point: The ZRF data isn't accurate enough to tell us whether Einstein's equation (1) or Newton's equation (2) is a better description of reality. And Newton's version of gravity doesn't account for gravitational waves, which means that ZRF could not measure gravitational waves.
Gravitational waves are changes in the gravitational field that don't satisfy equation (2); they really require $c$ to be finite, and really require the full form of equation (1). The reason that GW150914 is important is because it is the first direct measurement of a gravitational field where equation (2) really is not enough; for GW150914, $c$ must be finite, and equation (2) is simply wrong. Even the gravitational field of a single black hole doesn't vary in time, so that $\partial^2/\partial t^2$ side is just $0$ [though there are things I left out of equation (1) that come in]. Seeing the dynamic (time-dependent) behavior of the gravitational field in this form is a big deal. ZRF did not do this; their data were consistent with both finite $c$ and with $c=\infty$.
Conclusion
Finally, if you still don't believe me, here's a sociological argument: Just look at the paper by Zumberge, Rinker, and Faller as cited by the OP. The only time the word "wave" is used by ZRF is when referring to the wavelength of their laser. Gravitational waves were predicted in 1916 and the topic of active research at the time ZRF published. So not only are they not being given credit for "discovering gravitational waves", they themselves did not claim to have done so — presumably because they knew that they had not. The effect that ZRF measured was varying in time primarily because the location of their measuring device was changing. Although the relative positions of the sun, earth, and moon were also changing, that was on a much slower time scale, such that what they saw was consistent with Newtonian gravity, and was not a detection of gravitational waves.
You can disagree about the definition of the phrase "gravitational waves" with ZRF, LIGO, the rest of the physics community, and everyone here on stackexchange all you want, but what matters ultimately is that LIGO has found something truly unique in the history of science.
This is the SAME gravitational wave effect measured by the LIGO researches recently
It is not. Bob already gave a nice answer, but I would like to add a couple of layman analogies.
Picture a pond. With simple instruments you can measure the water level over seasons and you will get some wavy trend like low in summer, high in winter, however this is just the water trend. Measuring water waves means being able to detect the tiny ripples on the surface when a drop falls into the pond, and this is a totally different story.
Considering air instead of water, the analogy consists in detecting the atmospheric pressure, easily done with a barometer, compared with the recording of sound, for which you need at least a phonograph. Note that between these two instruments there are more than two centuries of physics and engineering research!