As a climber who is also scared of heights, I like to describe it as "a rational fear of heights". If I'm actually safe, I'm fine with it. It's the thought that a single slip, mental or physical = death. Even with a waist-high barrier, I'm still not going to look over the edge for fear of leaning too far. Give me a glass room hanging over the grand canyon that's stable and I'll be happy to go inside, lie face down on the floor, etc.

That is a rational fear. What you are describing is a normal fear. In my caseit would be more of a phobia where even when I am perfectly safe I would sweat, heartbeat would increase and I might get into a panic attack as suck.