This is really a comment rather than an answer, but it's not going to fit in the comment box. I haven't yet watched the video, so I am taking your quotes at face value.
I will say that the Veritasium channel has in my experience had really excellent research, and their clickbaity "everyone is wrong about X" videos tend to be correct explanations about differences between popular science and actual research that people tend to slowly realize over a year or so in graduate school. That kind of communication is hard, because you have to (a) explain what a person qualified for graduate school might understand, then (b) explain how that understanding might be slowly modified over a year or two of study of related topics. The fraction of this journey that a person will make during a forty-minute video depends a lot on the viewer's starting point and their individual acumen. The starting point that "the video" as an entire object is "right" or "wrong" is itself a misunderstanding of what's being communicated.
To your questions:
However, as I understand it, Quantum Mechanics is a local theory, or at least it isn't necessarily non-local.
I don't have a lot of confidence that everyone involved in a discussion will come to the table using all of these technical terms the same way.
When most people say "quantum mechanics," they mean the model based on the Schrödinger eigenfunction equation,
$$ \left( \frac{\hat p^2}{2m} + \hat V(x) \right)\psi(x,t) = \hat E\psi(x,t), $$
where the momentum and energy operators $\hat p = -i\hbar\partial_x$ and $\hat E=+i\hbar\partial_t$ mean that a plane-wave state $\psi = e^{i(kx-\omega t)}$ has momentum eigenvalue $\hbar k$ and energy eigenvalue $\hbar \omega$.
Note that the Schrödinger equation is explicitly nonrelativistic. The relativistic energy for a free particle obeys $E^2 = (pc)^2 + (mc^2)^2$, which approaches the Newtonian $E = mc^2 + (pc)^2 / 2mc^2$ only in the low-momentum limit $pc\ll mc^2$. (The constant offset $mc^2$ turns out to be irrelevant here.) So at some level, tension between the Schrödinger equation and relativity is to be expected. Consider that Dirac's development of a relativistic wave equation immediately predicted both electron spin and antimatter electrons.
Contrary to your understanding, solutions to the Schrödinger equation are explicitly nonlocal functions. Functions of complex numbers which solve differential equations tend to be analytic, which means (among other things) that you can't change the behavior of a function in one place without changing it in some way everywhere. Students in introductory quantum courses will solve lots of "boundary value problems," where you demand that the wavefunction or its derivatives have particular values at one place, and combine those constraints with your potential function to solve for the wavefunction everywhere.
In the context of the video, it also seems to imply that reality must be non-local. I thought it was not locally real.
I think there's also a problem with your technical terms here. I think the correct replacement for "it" in your second sentence would be "reality is not locally real," which sounds unnecessarily bizarre.
In any case, we don't know what reality "is." We know what this or that model of reality is, and whether those models describe phenomena that we can observe. It is often the case that our best models of some microscopic phenomenon have behaviors or internal features which are quite different from the intuition that we developed from observing macroscopic phenomena. But a statement like "our model which is X is good, therefore reality is X" is fraught. To quote a cliché, the map is not the territory.
However, I thought the main point of Bell's theorem was that it does rule out local hidden variable theories.
Careful here. Bell's theorem doesn't rule out anything. Bell observes that, if two particles carry their own state information away with them to separate detectors, you get a particular statistical distribution of the correlation between their measurements. If the state information is shared or "entangled" between both particles, even as they separate, you'll get a different distribution of correlations. The theorem is that the quantum-mechanical superposition gives the largest possible correlations.
It is experiments which have observed, consistently, that it's possible to create ensembles of particle pairs whose measurement correlations follow the distribution for maximally-entangled particles. In low-noise experiments, it's possible to exclude a large class of models where the state information travels with the particles. The Nobel Prize for entanglement went to the experimentalists Aspect, Clauser, and Zeilinger. (If Bell were alive he would certainly have been included; he would have turned 98 this year.)
This is also stated on the Wikipedia page of Bell's theorem.
The amazing thing about Wikipedia — the thing that makes it trustworthy enough that people treat it incorrectly as an oracle — is that when it's wrong, you can click "edit" and fix it. The next sentence you quote says that this interpretation is present in "all sorts of physics textbooks and papers and whatnot." You found one.
I could not find a source on Bell stating that [it's really quite remarkable how many people make that error].
Bell discusses this at length in his book "Speakable and unspeakable in quantum mechanics" (1987).
Lastly, I was wondering if they could be using the definitions of locality, realism, and hidden-variables differently or ambiguously. Are these statements not wrong in terms of the definitions used by physicists?
If you're not sure that everyone is using the same terms, the end of the question is an inefficient place to raise that concern.
Bell's 1964 paper is much easier to find now than it used to be. (When I went down my Bell's Theorem rabbit hole as a grad student twenty years ago, I had to use secret library jujitsu to find a copy; the journal that published it failed after about a year.) If you have questions about how things are defined, going backwards to original sources is generally the way to go.
I think I have been explicit above about how I understand the distinction between a "local" state variable, associated with an individual particle, versus a "nonlocal" state variable, associated collectively with the pair.
The question of "realism" has to do with whether the result of the measurement exists before it is measured. I have a bowl of fruit in my kitchen that has some apples in it. I don't know how many are left; it is either two, three, or four. But I don't believe that the number of actual apples in my actual bowl is ambiguous, nor that the wavefunction of the bowl will collapse into e.g. the three-apple state when I go to look at it, nor that I can keep the number of apples ambiguous by pulling one off the top while carefully not looking at the bottom of the bowl. All of these counterintuitive statements are things people do say about quantum-mechanical states. These ideas are particularly associated with the "Copenhagen interpretation" of quantum mechanics, in which a special process called a "measurement" produces an instantaneous global change in a wavefunction.
The idea of a "hidden variable" (which originated somewhere along the line from EPR to Bohm to Bell) is that, while an observable like spin certainly exhibits all of the Copenhagen non-realism, perhaps the spin is actually related to some other state information which isn't experimentally accessible.
I did wind up watching the video while I was writing this. I had apparently watched the first half earlier and stopped, so I didn't go back to the beginning, but your questions are all about the ending. The video's statements at the end about what has or hasn't been proved are all consistent with my understanding, but the point they are making is pretty subtle, and I don't think your question summarizes it correctly. As I wrote at the top of this answer, I think it's fine to sit with the idea that your previous understanding is different from the actual state of things, even if you can't yet summarize the actual state of things to your own satisfaction.