Well, using infinite set would be better either way as this is a property of all infinite sets, not just countable (infinite) sets. It would also avoid any confusion due to this difference of convention.

That said, I didn't actually know about your convention*, so thank you for making me aware of it.

*probably because any easy way of saying "at most countable" in my first language would be confused with "very countable" or even "mostly countable", so I'm guessing none of my professors thought about/remembered this other convention when defining countable sets.