Term for altitude of the sun?
The sun travels in an arc across the sky. Where I live, during the summer the arc rises higher, but in the winter the arc is lower. Once might speak of highest altitude or elevation the sun reaches during any given day. In the old days this was called "high noon" when the sun had reached the highest point in the sky for that day. The town clock would be set to 12 at this point.
Is there a term for the elevation of the sun in the sky?
The "arc" you talk about is called the Ecliptic (https://en.wikipedia.org/wiki/Ecliptic). This is the path the sun takes in the sky, and would stay stationary if the earth weren't tilted, but as it is tilted the suns path in the sky varies over a year giving us the seasons.
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.
Astronomers use the term "elevation" as a synonym for "altitude". Some examples: https://dept.astro.lsa.umich.edu/ugactivities/Labs/outdoor_series/constellations/constellations.html , http://www.astro.caltech.edu/~mcs/CBI/pointing/ , http://www.gb.nrao.edu/GBTopsdocs/primer/horizon_coordinate_system.htm .
Actually is it always at the highest point when it crosses the meridian? Eg- In the Northern Hemisphere, between mid-March and mid-June might it be slightly higher a few fractions of a second afterwards? (Genuine question, not just being an annoying pedant)
@adrianmcmenamin Good question. I thought the solar noon was defined to be the moment of crossing the meridian. Maybe the sun is slightly higher a few seconds before or after because of the analemma. https://en.wikipedia.org/wiki/Analemma The analemma shows us the sun's noon position changes a bit from day to day, making a slender figure 8 in the sky.
@adrianmcmenamin One might imagine an exception for when somebody is standing at the North Pole on the longest day of the year. Then the sun just goes in a circle above the horizon each day. That's the one time and place (in the Northern Hemisphere) where the sun is always at the highest altitude for every minute of the day. Even then, it is at its highest when it crosses the meridian simply because it is always the same altitude for every minute of the day.
@adrianmcmenamin http://astronomy.stackexchange.com/questions/937 addresses the question of declination change being fast (for the moon, not the Sun) enough to alter the highest point. Could you ask your comment question as a separate question? I think it might be interesting to see how many seconds it really is.
Actually, http://astronomy.stackexchange.com/questions/1213/ already asks this other question.