the sun - Term for altitude of the sun?

The term for when the sun is at its highest is "solar noon".

https://en.wikipedia.org/wiki/Noon#Solar_noon It is the moment the sun crosses the meridian. The meridian is an imaginary half-circle that goes from the northernmost point on the horizon to the highest point directly above you and then to the southernmost point on the horizon. The sun is always at its highest point in its daily path when it crosses the merdian.

You're right that in earlier centuries, each town would set its clocks at noon for when the sun was at its highest point. Those of us who are familiar with astronomy and history would call that type of timekeeping "apparent solar time". You can't guarantee an exact 24 hour day with "apparent solar time", so some people used "mean solar time" instead. (Historical note: Having each town choose its local noon became a nightmare for railway companies that wanted to set train schedules. So the railway companies urged the creation of 4 time zones in America and encouraged each small town to use "standard time" instead of "apparent solar time".

And as rgettman already mentioned, the term for the sun's height above the horizon is "altitude". Astronomers use the term "elevation" for a person's distance above sea level.