I'm a pilot and I'm terrified of heights. I get vertigo even thinking about looking over the edge of a cliff. I would also really like to try skydiving one of these years.

I think that a better way to express it for many people is we're not afraid of heights, we're afraid of falling and dying. Removing the falling aspect "cures" the phobia.

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.