General is applicable to more circumstances than special. For example, \(E=mc^2\) is a special case of \(E^2= (mc^2)^2+(pc)^2\) because the former is only valid when \(p=0\), or when you are considering an object at rest. The latter is valid for all p.
I think you nailed it. Let me just add that general and special are also relative, i.e., a general thing can be a special case of an even more general stuff.
The General Theory deals with the universe at large: The operations of curvature and mass (planatery mass and orbitals), and even singularities, whereas the Special Theory deals with moving frames of existence, where two observers are necessery. They are simply ideal names.
The Special Theory of Relativity works only when we can completely ignore the affects of gravity. As such it is great for small and fast systems. The General Theory of Relativity is actually general, working in all the cases where the special theory works as well as when gravity cannot or should not be ignored. A proof of a mathematical statement, say "There are no non-trivial solutions in positive integers for \(a^n + b^n = c^n\)" would be very special if n was limited to just one number like 3. But it would be more general if it were for n equal to all multiples of 3 or all odd primes or to all values of n greater than 2.
Another difference is that the mathematics that underlies special relativity is understandable to even high school students while the mathematics that underlies general relativity is quite esoteric. General relativity is a lower level graduate-level class at most colleges. Nothing. There is no viable theory of quantum gravity (yet).
