How about replies from native speakers only? (hight? Really?????).

As to the query itself, this is something that has bothered me for years. No amount of research has resulted in a satisfactory conclusion. In the specific context given, my most regular client insists on 'height profile'; but for hills etc the same client insists on 'elevation', and for anything above ground level it has to be 'altitude'. Other clients - and some texts I find here and there as reference - reverse that preference. Most of my colleagues seem as puzzled as I am.

I would personally not use "elevation profile" in AE, unless the context is architectural. An "elevation" has a completely different architectural meaning. "altitude profile" would be good in AE for something like a GPS track. "Flight profile" might work for an airplane.

height above a given level, especially sea level.

"the area has a topography that ranges from 1,500 to 3,000 metres in elevation"

Best: elevation, except maybe in architectural drawings, where the term is used differently.

Maybe: altitude, especially in regard to airplane flights.

Not: Height, except maybe in some very specific industry or context.

There's nothing wrong with the word altitude, including as used in #12 of course. #11, however, refers to words coupled with profile. Many uses of altitude have an element of reference to how thin the atmosphere (oxygen) is (very relevant for athletes and those using non-turbocharged combustion engines). I don't think "altitude profile" is the first choice for most purposes, but might expect it for those who hike or climb at high altitude.

Maps often show elevation (above mean sea level) of the earth/ground, and maybe structures on the ground. So, for example, one could find the elevation of a high-altitude ski resort and an elevation profile of the various peaks.

Especially in a context where ground elevation is given, I can imagine use of height profile to show the height of the tree canopy or of buildings or other structures above the ground elevation (i.e., the delta), in cases where the exact peak elevation of the top above sea-level and the amount of oxygen available are not the concern.

From my limited exposure to such things, I would have thought that the term used for architectural and engineering drawings was "elevation" not "elevation profile".

Shame on me: I'm blushing.. height of course.

#14: Are we permitted to know just what your Muttersprache is - assuming you have one?

#13: Yes, fair enough.