Entropy is...disorder?
I personally find the terms consistent. Think of the entropy as Boltzman proposes: $S=k \, \ln W$ Meaning high entropy states can be realized via many different configurations. Truly ordered state (assume you arrange a sculpture from atoms) can be realized via much smaller number of microscopic states. So again, equilibrium is not order - it is a mess.
What you are missing is the microscopic definition of entropy, once you know that, you will understand why people say that entropy is disorder.
Equilibrium as order
First, let's address your valid intuition that equilibrium as a form of order. Indeed, if everything is in thermal equilibrium, you just need to measure the temperature somewhere, and then you will know the temperature of everything. In our out of equilibrium, my body, my laptop, the room, outer space, all have different temperatures, and I need more information to know the state of everything, and I feel this is less "ordered" than the thermal equilibrium case.
What transpires is that less information needed corresponds to a higher degree of order. Well, let's keep that in mind for the next bit.
Entropy is microscopic disorder
In Physics, we know that the properties of macroscopic objects are determined by the motions of the particles that compose them. In particular, temperature of a gas is the disorganised jiggling of the atoms making it up.
As you increase the temperature, the atoms will move more and more erratically, and will have diverse speeds at any given time.
As you cool it, the particles will move slower and slower, until perhaps they freeze in place, forming a solid.
Which of the two - the still, regular lattice of the solid or the whizzing commotion of the particles that forms a gas - seems to you more disordered? Definitely the second. You know from thermodynamics that the gas has higher entropy than the solid. Indeed, there is a precise formula linking the macroscopic state variable $S$, entropy, and the microscopic conception of disorder I described.
Conclusion: the two ideas are reconcilable
In the projected "heat death" of the universe, everywhere there is constant temperature and density. In that sense, the universe is homogeneous and thus ordered. But microscopically - in the movements of the particles - that is the state in which there is the least order: no structure whatsoever, just a big soup of whizzing particles.
First of all as stated by Madan Ivan: equilibrium is not order. But you can get certain systems that are in a meta-stable "local" equilibrium (here meaning that you need some energy to move it from there), for example a crystal. These can be highly ordered.
Intuitively: if you smack the crystal with a hammer it breaks to pieces. This brings your closer to the global equilibrium. In the universe as a whole there is energy exchange between such subsystems and the second law of thermodynamics states that the overall order decreases by these processes.
So I think your problem is the two uses of the word equilibrium. Meta-stable equilibria can be order while the one that is used in the second law is the global minimum.
A comment on entropy in general: there isn't just one, there is a lot of them. In thermodynamics only there are 3 distinct ones. The names I use in the following are not official, since the literature mostly does not distinguish between them.
- The Gibbs entropy: $$S_G = -k \sum_{N} \int d \tau_N p_N \log(p_N) $$ where the sum is over all the states of the system and $p_N$ is the probability of it. It turns out that this is a constant of the equations of motion.
- The Boltzmann entropy: $$S_B = -k \sum_{1} \int d \tau_1 p_1 \log(p_1) $$ where $p_1$ is now the one particle distribution. This entropy is just wrong, but used a lot.
- Experimental entropy: $$\Delta S_E = \int dQ/T $$ This is the one that increases.
It can be shown that both 1. and 3. are important quantities, but the second law applies to the 3. one.
References: Unfortunately I can only link to this http://www.oxfordmartin.ox.ac.uk/event/1348 which is where I got the information from.