It looks like (if I've parsed right) every one of them stands for "Earth", except that HEO alone can also be overloaded three ways (high-earth, highly-elliptical, and highly-eccentric).
This is unimportant, but: a site:nasa.gov search shows all three "HEO" acronyms in common use, there; and even Wikipedia abbreviates it inconsistently across entries[0-2].
The phrase "highly elliptical" is one where I know exactly what they mean but the more I think about it the more wrong it seems. It should be "Highly eccentric orbit".
All shapes which satisfy {(x,y)| x^2/a^2 + y^2/b^2 = 1} for fixed values of a,b in R are elliptical. Something is either elliptical or not - it's not a matter of degree. A circle is just as elliptical as a more eccentric ellipse in the same way that a square is just as rectangular as a more elongated rectangle.
I thought GEO stood for Geostationary Earth Orbit, since a geostationary orbit must be equatorial anyway. But actually "Earth" would also be redundant, since "Geo-" already stands for Earth.
I understand it's both, but "equatorial" is more precise to distinguish it from GSO, a non-equatorial [g]eo[s]ynchronous [o]rbit. Otherwise, they would both be "GEO".