Pet peeve time. This is a button-down, because the collar buttons down:
This, despite having buttons down the front, is not a button-down:
I know I’ve already lost this one — and I’d go so far as to say that “button-down” simply means something else to your average under-30 — but it’s a useful distinction and I’m not letting go.
(And if you’re about to say “Fine, what should we call them?”: Both are dress shirts, one with a button-down collar and one with a point collar. You can probably even call the second one just a “dress shirt” or even just a “shirt” if you’re not in company that means “t-shirt” when they say “shirt”. The ones that aren’t dress shirts but which still have buttons down the front are sport shirts, with a similar etymology to “sports jacket”.)