When an electron changes its spin, or any other intrinsic property, is it still the same electron?

It doesn't matter.

Suppose two electrons approach each other, exchange a photon, and leave with different spins. Are these "the same electrons" as before? This question doesn't have a well-defined answer. You started with some state of the electron quantum field and now have a different one; whether some parts of it are the "same" as before are really up to how you define the word "same". Absolutely nothing within the theory itself cares about this distinction.

When people talk about physics to other people, they use words in order to communicate effectively. If you took a hardline stance where any change whatsoever produced a "different" electron, then it would be very difficult to talk about low-energy physics. For example, you couldn't say that one atom transferred an electron to another, because it wouldn't be the "same" electron anymore. But if you said that electron identity was always persistent, it would be difficult to talk about very high-energy physics, where electrons are freely created and destroyed. So the word "same" may be used differently in different contexts, but it doesn't actually matter. The word is a tool to describe the theory, not the theory itself.

As a general comment: you've asked a lot of questions about how words are used in physics, where you take various quotes from across this site out of context and point out that they use words slightly differently. While I appreciate that you're doing this carefully, it's not effective by itself -- it's better to learn the mathematical theory that these words are about. Mathematics is just another language, but it's a very precise one, and that precision is just what you need when studying something as difficult as quantum mechanics.

Another question, which I think you implied in your (many) questions, is: under what circumstances are excitations related by changes in intrinsic properties called the same particle? Spin up and spin down electrons are related by rotations in physical space. But protons and neutrons can be thought of as excitations of the "nucleon" field, which are related by rotations in "isospin space". That is, a proton is just an "isospin up nucleon" and the neutron is "isospin down", and the two can interconvert by emitting leptons. So why do we give them different names?

Again, at the level of the theory, there's no actual difference. You can package up the proton and neutron fields into a nucleon field, which is as simple as defining $\Psi(x) = (p(x), n(x))$, but the physical content of the theory doesn't change. Whether we think of $\Psi$ as describing one kind of particle or two depends on the context. It may be useful to work in terms of $\Psi$ when doing high-energy hadron physics, but it's useful to work in terms of $p$ and $n$ when doing nuclear physics, where the difference between them is important.

It always comes down to what is useful in the particular problem you're studying, which can be influenced by which symmetries are broken, what perturbations apply, what is approximately conserved by the dynamics, and so on. It's just a name, anyway.

Spin is a complicated quantity in quantum mechanics. If you want to really understand it, there is absolutely no substitute for a complete reading of a full-grown textbook. (That means: Cohen-Tannoudji, Shankar, Sakurai, or equivalent level. Introductory textbooks like Griffiths are OK as an on-ramp, but not the full deal.)

Spin is complicated because it is

  1. an operator quantity, i.e. a quantity that need not have a well-defined value;
  2. a vector quantity, i.e. a quantity with three independent compontents; and moreover
  3. a vector operator whose components are incompatible (i.e. do not commute) with each other, which means that if one component of the spin has a well-defined value, then the other two will not.

This means that the spin comes with three components, $\hat{S}_x$, $\hat{S}_y$ and $\hat{S}_z$, but only one of the three can have a well-defined value at any given time.* However, that said, there is one more relevant quantity, which is the total spin, i.e. the combination $$ \hat{S}^2 = \hat{S}^2_x + \hat{S}^2_y + \hat{S}^2_z, $$ which commutes with all of the individual components, and that means that the most complete set of information you can get about a system with angular momentum in three dimensions is the total spin, $S^2$, and one of the components (traditionally taken to be $S_z$, but it is crucial to emphasize that this can be along any direction you might care to choose).

Moreover, because of technical reasons to do with quantization, the possible values of these components are restricted: the total spin can only take values of the form $S^2 = \hbar^2 s(s+1)$, for $s\in \tfrac12 \mathbb N = \{0,\frac12,1,\frac32,2,\ldots\}$ a nonnegative integer or half-integer, and the total spin projection can only take the values $S_z = -\hbar s, -\hbar (s-1), \ldots, \hbar (s-1), \hbar s$. When we say that a given system "has spin $s$", what we really mean is that it has total spin $S^2 = \hbar^2 s(s+1)$.

For electrons, these two quantities play very different roles.

  • The total spin is intrinsic. All electrons have total spin quantum number $s=1/2$, which means that they have total spin $S^2 = \frac34\hbar^2 $, and nothing you can do to an electron will change this.
  • The spin projection, $S_z$, on the other hand, is not intrinsic, and it basically tells you which direction (within the bounds of the quantization of angular momentum) the spin is pointing.

When you do things like spin flips with an electron, you're changing the latter, not the former.

$\ $

* With one exception when they're all zero, with total spin zero.

What people mean when they say that spin is an intrinsic property is that spin represents an internal state of the particle that exists independently of its position and motion in space. However, the value* of that internal state can and does change, and when that happens that doesn’t mean the electron can be meaningfully said to have been replaced by a “different” electron, any more than an electron that changed its position in space would be thought of as a “new” or “different” electron. We just say that the electron moved.

Similarly, there is nothing strange or inconsistent about thinking that the spin of the electron changed, and there is no need to explain the strangeness away by saying the electron has been replaced by ”another” electron. A change of spin is a completely reasonable thing to imagine, once one has overcome the small hurdle of understanding what it means for spin to be “intrinsic”. It is not the particular direction in space of the spin that is intrinsic, rather, what is intrinsic is the set of labels that spin can assume (that is, the vector space - $\mathbb{C}^2$ in the case of the electron - where spin “lives”) along with the precise rules that govern how the spin internal state evolves and interacts with position and other parameters of the quantum system.

* Another subtle issue here is that one usually cannot consistently talk about the spin having a value in the sense of a particular direction in space that the spin vector is “pointing”. This is the difficulty alluded to in @EmilioPisanty’s answer, having to do with the fact that the three coordinates of the spin operator-valued vector do not commute, which means they cannot simultaneously be thought of as having well-defined values. This issue is tangential to my remarks above, but still important to mention, as it illustrates another way in which the words that physicists use to talk about ideas in physics fail to communicate nuances of meaning that can only properly be conveyed using precise mathematical language. As @knzhou says, in order to properly understand what spin is, there is no susbstitute to learning the mathematics behind it.