Lost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct from a torus (genus 1) and no homotopy exists between them.
Clearly IANAM.
Clearly IANAM.
A torus is like an inner tube - an inner void and a big hole in the middle.
A solid torus just has a big hole in the middle, like a donut.
Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?
Another way to consider it is if you have a circle around the valve hole, you can't stretch that circle to be one of the two possible circles you could draw on a torus (around the 'trunk' and around centred on the donut-hole).
Anyway, I think they just chose a bad example, this was just where my mind went.