actually i like to be proven wrong because thats how you learn.
in this case, however, please look at some primary literature and you will quickly find that "annealing" is a synonym for "heating" in metallurgy, be it to 100°c to decrease dislocation density (and maybe first precipitation of some transition carbides) or 1200°c for homogenising higher alloyed steel. the temperature in question is stated. if you think about it, this makes more sence than using different expressions (like some text books do) because they create a false illusion of dealing with some separate, well defined phenomena. e.g. if you understand "annealing" as you state above you still dont know if its spherodized of a det anneal. besides there are about dozen processes called "xxx annealing" as you must be well aware.
so to make it short, imo the use of "anneal to xxx°" is a more precise and efficient expression.