Firstly, I assume that when choosing between two moves they will absolutely never choose one that they discern is a forced checkmate against them regardless of the unlikelihood that a human opponent will find it, as long as the alternative does not result in that forced checkmate as far as they can see. (I'm guessing that human grandmasters might deviate from this algorithm, hoping the opponent will miss the checkmate, because the other move sucks. Correct me if I am wrong.)

But my question is this:

While there is usually a theoretically "correct" gto move that the computer is usually finding, what about when it is already in the midst of a situation where it knows that a perfect opponent will checkmate it. If it is asked to play on is there still some sort of gto type criteria in the hope that the opponent messes up? Or in this situation, do the various computers have major differences in how they choose their moves.

I think these types of questions have to do with catching up on your knowledge of modern chess engines and the humanity of them. We're going to assume these are engines that are playing at the top of their strength.

That is correct. There may be some positions the engine won't quite understand , but it's rare for sure.

If it's choosing between a move that leads to mate in 7 and a move that leads to mate in 8, it depends on the settings of the engine but they'll still play the "best" move for their position. A move that leads to mate in 8 is better than a move that leads to mate in 7.

Human players, especially strong ones, will most definitely play a complicated losing position than a simple winning one. You should never play "hope chess", though. (IMs and GMs will even play a completely lost position for stalemate tricks, sometimes.)

I really would not think of this in terms of GTO. GTO is not a concept in chess. Yes, engines can play the "optimal" move but it's not really a term used in the chess world. I don't know about it "knowing a perfect opponent will checkmate it" because it's a chess engine that's supposed to win. The major engines have differences in how they play their moves, yes. Sometimes engines will make "waiting moves" or moves that don't make a ton of sense on the surface to human players, and when analyzing or faced with one of those types of moves it's confusing and/or strange. A random king move, for example, when the position doesn't call for it. There have been breakthroughs in the last few years with neural network engines that play a more human-like style. I think you should read up on those, because that's essentially what you're asking for.

Leela played itself millions of times to get an understanding of chess. Maia is an engine that was developed through only human-played games. Maybe read up on that?

They're not supercomputers anymore. Chess engines are AI. The most advanced one (AlphaZero) is based on neural networking. They search tens of thousands of positions in a second.

It's hard to answer your question because of stuff like this

https://www.youtube.com/watch?v=w2BWmSBog_0&vl=en-US

There was another blunder by one of the more advanced chess engines within the past week or two that was on the chess subreddit, but I cannot find it. They're intelligence apparently is capable of replicating human error.

AFAIK, chess computers don't take into account "practical" considerations. It will simply play the line that produces the highest evaluation, even if it's dead losing. I don't think computers ever go for traps hoping the opponent will make a mistake (someone can CMIIW). GMs also are never going to play a move that would allow their opponent to checkmate them if they see it. "Don't play hope chess," as the saying goes. GMs will absolutely take less-than-ideal lines just to get their opponents out of opening theory. Magnus Carlsen has routinely talked about this, figuring that his advantage in calculating and understanding complex positions is ultimately going to matter much more than getting a slightly worse position out of openings that have been studied to death.

I don't think this is quite true. If a GM is in a position and they see two paths:

Path 1: allows a multi piece sacrifice that is highly unintuitive, but if seen will allow checkmate in 10 moves

or

Path 2: will result in a fairly straightforward probably losing position with some chances of scraping a draw

I think many GMs might go for Path 1. I believe every computer would choose path 2.

With the qualifier that there are hundreds of strong engines out there, and we don't really know what's going on internally in most of them, from what I've seen the answer is basically NO, computers don't employ any kind of GTO-type considerations when they're losing. They more or less play in a way to prolong the game as long as possible, even if it makes the opponent's win simpler. And I think even that is only an indirect result of the computer following it's evaluation algorithm to the letter. (to put it simply, a move that gets checkmated instantly is rated worse than a move that gets checkmated further down the line)

The thing with chess engine development these days is that engines left humans in the dust long, long ago. So basically all engine development that is going on currently is only in service of competition with other chess engines. So in that sense, there is no reason for the programmers to pay any mind to GTO-type strategies (GTO is not exactly correct but we know what you mean). Their silicon opponents are going to see the best moves, so, there's really no need to spend resources on the contingency of them failing to do so