The general question has merit, but I feel like it's hard to come to an all-encompassing answer, so I'll reply with my take for your specific example.
Like, for instance, would you still call a spider-centaur a Drider, or would you prefer some other name? Would any other change be warranted, like maybe they'd be more like Quelaag, having an entire spider as their lower half (including mouth bits), and/or an elemental shift?
Since drider is literally a portmanteau of drow and spider, I'm guessing there'd be an ever stronger reason to change it than with tieflings and halflings. Humder would still be a bit on the nose, though you could go with Ummer, or choose another source for the name.
The two mouths bit I think is a Watsonian matter over a Doylist one: I wouldn't feel the need to change it just for resemble the original creature less, but there could be a chance to add interesting spins to the concept. Example: The human part is an appendage for the spidery main body to manipulate objects and maybe bait humans; as such, the human "half" lacks a brain and their eyes and mouth are mostly vestigial and crude-looking (if even present), so the spider mouth and eyes would be the real ones. Extra creepy on a visual level, and opens the road for truly inhuman characters. Two functional mouths could also work if the race is supposed to be a heavily magical one, if it's closely related to the Fae in-universe for example.