The terms are used so interchangeably by people that I'm not surprised you might disagree. I looked at a bunch of definitions and tried to distill a useful distinction.

Given that people hire coaches, but usually not mentors, the distinction for me is that a coach is engaged towards a goal and therefore is more directive of what you need to do to get there.

I don't think your example question is a great one for exploring the distinction because it's a single, binary question. But even in your example, despite the coach responding with questions, you describe the coach as pushing you to go do some work: figure out the differences yourself, or come up with a project to explore the question.

In that sense, I see the coach as "setting the agenda" whereas the mentor is having a more open-ended conversation about it.