You are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation.
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?
My understanding is they're non-homeomorphic. You can thread a string in through the inflation hole (valve hole) on an inner tube, around the great circle inside the tube, and then out the valve hole. You can also, of course go through the centre as with a donut/bagel/torus. But you can't do the former threading with a donut, so the torus and whatever the inner-tube are not typologically equivalent.
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.
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?