Or am I misinterpreting the meaning of all the different version strings?
I think that you are (if miners are using completely different code to Bitcoin Core then please give me the github link to their source).
I don't know really. But I do know that in the recent past, it became obvious that some miners were, for example, mining without verification (SPV), which suggests that they were not mining to any Bitcoin Core s/w, right?
I think 'completely different' is unlikely, in any case. But you could change a lot of stuff and still not be 'completely different'.