Argh! Classifying things is difficult. These are my thoughts, and I present them as such - not facts. Feel free to disagree.
A simple description of power metal is just fast paced heavy metal with a focus on melodic instrumental work, and usually with clean vocals. By massively generalising, I'd say there are two main types of power metal: heavy and melodic. That's not to say heavy power metal bands are unmelodic, just melodic ones focus far more on melody than heaviness.
Heavy power metal is seen in bands like Rage and Blind Guardian.
Melodic power metal is seen in bands like Rhapsody and Dragonforce.
Sadly, there are now lots of unoriginal super, hyper, ultra, epic, symphonic, film-score, fantasy power metal bands these days (or whatever the trendy description is these days) who try and do exactly the same thing as Rhapsody and Dragonforce, but much worse. This doesn't do much in terms of credibility for a genre that isn't taken very seriously by most anyway. The other disturbing thing is that at a Dragonforce gig I went to earlier this year, the majority of people there must have been under 18 and treated the support act (the one before Angra) with such immature disrespect that it was embarassing to be in that audience! I expect that from nu-metal bands, but it shocked me that a band that was relatively underground until recently and actually has musicians in could attract such a moronic fanbase...
There's also progressive power metal. Kamelot are progressive power metal in that they write catchy melodic music but balance it with more progressive musical ideas and themes. Angra are also progressive power. The progressive and power metal ideas seem to be balanced fairly evenly with this sort of music.
I suppose you could classify Symphony X similarly, but I really don't think Symphony X should be called a power metal band. I listen to power metal for the fast paced and energetic nature of the music, whereas Symphony X more because I like to listen to the music, lyrics - everything. I'd say Symphony X were progressive, with some power metal aspects, and formerly neo-classical ideas.
In addition, you have bands like Bodom who could be described as borderline power, borderline death.