We NEVER wear shoes inside our homes in Macedonia, but going to the gym I was required to bring "indoor shoes/sneakers". Ie. ones that I will use exclusively there or indoor environments. Helps keep the gym cleaner.

I guess what I'm saying is, it's not an American thing, it's a gym thing.

What kind of shoes are you allowed to wear in a gym in Canada?

I think it's an language thing. Shoes I would wear for indoor activities like the gym I'd generally refer to as "gym shoes". Any other form of indoor shoes I'd just refer to as shoes. Indoor shoes to me immediately brings up inside the house, especially because that's how OP used "Indoor shoes" in context

Obviously people wear shoes indoors outside of the home. (The norm for a gym would presumably be a pair of sneakers.)

In the US, my experience is that people will sometimes/often take off shoes when they come into a home especially if they're going to track in snow, mud, etc. But it's certainly nothing like a universal rule and probably isn't even the norm.

Concept of indoor sport shoes is universal, because it is stupid to wear outdoor shoes for sporting indoors.