Interesting. When I have sized a wedding ring (something I have done several times because of a combination of not fitting, and of me losing a ring twice over the past 28 years) I obviously had to work to get the right size. Like you, I chose rounded edges which made it easier for me to get the ring on and off my finger. Originally I had a width of 4 mm, and moving to 3 mm worked much better for me. (My fingers are particularly narrow and long compared to most men’s hands.) But I have never been asked what “height” to use.

I would speculate that it affects the weight of the ring. (In my experience, too heavy a ring can be a problem. For instance, a heavy ring May vibrate against the steering wheel on long drives and make my fingers sore.) No one can really see the “height”, so why not go with the thinnest that makes the ring still sturdy?