Never checked but somehow got me surprised: aren't all chess engines deterministic? The same program, given the same input, should give the same output unless it is specifically programed to try to avoid it (and then we have PRNGs).
Some things that make chess engines non-deterministic:
1) Most chess engines use an opening book instead of calculating moves early in the game. These books are often programmed probabilistically, so it may have 4 reasonable responses to a given move and pick one randomly.
2) Many engines are programmed to "ponder", that is do optimistic calculations during the opponents turn. This means that the time taken by you to make the move could potentially affect the calculations performed by the engine.
You forgot two more things, they might not happen often enough because of the sheer number of possible positions in the early gameplay but it does make a difference in the game if you play the same position every time.
Many chess engines are (also) made for humans to play against.
Humans wouldn't want their opponent to play the exact same way every time. For a simple example, playing white, one wouldn't want the computer to always open 1. h4.
So, either the computer has to pick an opening, or the human opponent has to pick one.
Of course, that extends past the first move, so, assuming the computer to play deterministically, one would end up with the human opponent having to specify "let's play opening X, variant Y, as typically played by Z up to black's 15th move". It probably is easier to use a "set up a custom board" feature than to specify that.
And it wouldn't even stop there. The human specifying that particular opening likely wants to study that opening, not just the single continuation that the computer thinks to be the best one.
Also, as another poster said, a chess engine may use Monte Carlo simulation to pick a move, and that means using a random number generator.
A reply that was deleted mentioned that some engines use Monte Carlo simulations which are non-deterministic (?).
My experience with chess engines being non-deterministic is that if I pick the same opening I don't always play the same game with the same engine. I really might just be off here, and that variance might be related to random artificial handicapping.
Most chess engine opening books weight their branches to pick certain moves x% of the time, and offer 0 or negative value to 'bad moves' never to pick.
e.g. 1.e4, d4, c4 and Nf3 will get picked more often than 1. b3, c3, nc3 and f4, and 1. g3, e3 even less (or not at all).
Unless efforts are made for results to be deterministic, concurrent algorithms may complete their work / return their results in an ordering that is unstable (depending mostly on host OS and CPU management of time-slicing).
