2) "We apologize for any harm that we caused"
3) "We apologize for all the harm that we caused"
(1) is the least apologetic. This could be interpreted as saying "we might or might not have caused harm, and we think we didn't but you think we did, so we're going to apologize for your sake, but we're not really sorry because we didn't do anything wrong from our perspective".
(3) is the most apologetic. You acknowledge that you did something wrong, and that you're apologizing for it. This might be followed up with a specific list of the things that you did wrong, and that you are apologizing for. That would make for the most sincere apology. This could be interpreted as "we caused harm and we're sorry, whatever harm you think that we did, we agree that you are right, and we are sorry for all of it".
(2) is partway between (1) and (3). You acknowledge you caused harm, but you won't enumerate the things that you did wrong. So, you leave yourself a little bit of wiggle room. This could be interpreted as "we caused harm and we're sorry, but we didn't cause that much harm, we think it was actually quite little, you think it was a lot, but we're saying sorry, so let us go with this apology".
"We sincerely apologize for all the harm our research group did" would be a little less unnatural than "...for all harm...", but it's still an unusual choice of wording that ends up sounding like you want to emphasize that you did an unusually large amount of harm.
A more standard phrasing that includes the word all would be "We apologize for any and all harm...".
The standard form I know is actually "[I am deeply sorry] for any harm I may have caused..."; the letter here does not use a modalized verb. (Compare their "We sincerely apologize for any harm our research group did...".) In that sense it's more definite than usual. The criticism above is strange.
(The reason for the modality in the standard form isn't really to leave open the possibility that you didn't do any harm. It's to leave open the possibility that you did harm you don't even know about -- and therefore can't apologize for specifically.)
[1] In general, any occurs only in negative sentences, though in the details there are several types of sentences that are sort of "honorarily negative" for the purposes of allowing any and other words that obey the same restrictions.
It wouldn't have occurred to me to discuss it above, because I thought I was contrasting any with all, and neither is present in that example. But I'd rate it above most of the alternatives discussed (while still below "for any harm").
If you want to apologise, "all harm" admits you caused harm, which seems necessary for an apology. "Any" is a bit like those "I'm sorry if you were offended" non-apologies, though in this case the rest of the message does better than that.
I am surprised to see people hung up on this, because the rest of the letter acknowledges specific harms done. To me, it really seems like a minor thing, and something that I might have written (as a native English speaker.)