I think that the terminology is somewhat arbitrary here.
But here is my reasoning.
If your game starts with X then X, the second strike is
definitely part of the second frame, not the first. It
affects the scoring of the first frame, but it is not
part of it.
Similarly if you get a strike in the 10'th frame then you
need to throw 2 more balls to find your final score.
Because you are not going to throw any more frames, you
get the chance to throw them right away. But I would
like to classify them for scoring purposes as being no
more part of the tenth frame than the second strike was
part of the first frame. Certainly the scoring would
not be any different if the custom was that the extra
balls were thrown as part of an 11'th and 12'th (if needed)
frames which were not themselves scored.
So in terms of when they are thrown, sure. They are thrown
as part of the 10'th frame. But in terms of having a
simple consistent mental model of what the scoring means,
I find that it makes more sense to think of them as being
outside the first 10 frames.