No.
> And an endofunctor is just one of those where the argument type and return type are the same?
Yes - "endo" means that in general. E.g. endomorphism.
> Sometimes it makes me mad that so much confusion (and as another commenter put it, gatekeeping) has been sown in FP circles through the invention of pointlessly-obfuscated terminology for everything (and through - among many in those circles - a lack of interest in connecting the very simple dots that would allow masses of people to understand these concepts). It's like when the medieval Catholic Church refused to print a bible in anything but Latin so that they'd remain the sole authority on its content.
> Kudos to the OP for cutting through some of that, but shame on those who maintain this status quo.
> Edit: assuming it really is that simple. Some in the comments have suggested parts of the OP were wrong, so I'm curious to see discussion.
The precise terminology is vital; it's how we avoid these errors when talking about this very abstract stuff. Given that the OP is very wrong, precisely because of the kind of attitude you've taken here, I hope you'll take this opportunity to reconsider your views.