This may be too soft, and it may be a confusion only about formalism. But seeing why the homotopies equivalences I visualize in my head and that people describe are the same thing as the definition of a homotopy.
I can see why my visualization of a homotopy of maps is the same as the definition, since the definition basically says: "maps are homotopic is there is a movie taking one to the other." But, I have an issue with homotopy equivalences, since it seems I make the maps between the spaces the "movie" and not the homotopy between these maps and the identity.
When people describe why two spaces are homotopic, they seem to construct the same kind of "movie" that deforms one space into the other.
Take the following example. A sphere with a line from the North Pole to the south through the sphere. I think that this should be homotopy equivalent to $S^1 \vee S^2$. In my head, I visualize slowly moving the endpoints of the line through the sphere to one another, and once they reach each other, I have a sphere and a circle that touch in one point.
But, the definition of a homotopy equivalence is a map $f: X \to Y$ such that there is a map $g: Y \to X$, with $g f$ and $fg$ homotopic as maps to the identity.
If I try to see how my visualization would correspond to this, it would seem that the end result if the map $f$, and perhaps doing it backwards is the map $g$. But, what am I, and it seems most people I talk to, doing in the intermediate stages?
Please ask for clarification.