Does the wire “terminate” at the point where it is wrapped in the winder.
Well, not exactly.
The exact effect does depend on whether the wire is
bare or insulated, as well as how it is wound.
But you probably can find a length where the wire does
resonate on 40m that is somewhat close to the expected
value. There are a number of commercial versions that
work this way, as well as homemade versions using
plastic wind-up reels sold for camping clotheslines. I have
a version using a pair of mason's chalk line reels for this
purpose, but haven't had a chance to try them out yet.
The wound wire may act as an end-loading capacitance
(especially when bare wire is used), or as an inductance
(with insulated wire). While 80m / 40m should work pretty
well, the effects get less predictable as the amount of wire
wound up gets shorter. For example, winding the ends of
a 20m dipole to reach 17m is likely to require a different
antenna length than winding up an 80m dipole for the same
band.
So the wound wire still has an effect, but with a bit of
adjustment you can usually find a length that will work.