I think that describing infinity as an infinite process is bad because that way we are basically saying that this infinite process and infinity is the same thing!
I love ops explanation because it separates infinity (as a number) and infinite process (counting).
And I thought we are talking about kids? There are not many kids who would understand the professors way.
So isn't it first about making them understand the concept of infinity? That they can imagine it, before you bombard them with other abstract math concepts?