Bad example. I Googled the transcript, and they use the term "warp engines" three times in that episode before Sulu even says "The engines!"."Engines." My god, even Star Trek just called them that, like, a zillion times.
In "Tomorrow Is Yesterday," one of Sulu's lines was simply: "The engines!"
Should have gone with "Encounter at Farpoint". The term "warp engines" never appears in that entire episode. Heck, the word "engines" only appears once! (Wouldn't be surprised if TOS featured the term "warp engines" more that TNG at this point.)
Anyway, once you've said "Warp" half a dozen times in reference to velocity, you don't need the tell anybody which engines you're referring to. That doesn't mean they're not called "warp engines", it just means it's implied. I guess the Warp Scale is good for something after all.

Depends on the situation. If you're already in FTL, and someone says that the "engines" can't take much more of this, that's fine. So long as the audience can tell if you're referring to the FTL or sublight engines, it doesn't matter. However, as soon as you need to say that one is working and the other isn't, there needs to be a term for it, even if that term is just "FTL".I think, if the idea is to do a series, then you have one episode where the nature of the drive is described, and that's it. After that, "engines are offline," "core is offline" etc, will suffice for shorthand and avoid technospeak.