It's all up to opinion. When it first started, it was mainly a way of lashing out against all the Hair Metal bands in the 80's. So far, it's grown to include stuff like nu-metal and metalcore. Some people take it way too seriously; Trivium was "attacked" with balloons filled with urine by people shouting all that "Death to False Metal" crap. Personally, I don't like Trivium, but other people do, so why **** a show up for everyone else just to vent out your own prepubescent angst?
This is how I think of the two groups;
True Metal;
Heavy Metal
Thrash/Speed Metal
Power Metal
Black Metal
Death Metal
False Metal;
nu-metal
metalcore
grindcore
Funk metal
Industrial Metal
The genres of true metal are different from the other ones because the false metal genres were born from fusions and crossovers;
nu-metal; metal and rap/hip-hop
Metalcore; metal and Hardcore punk
grindcore; technically, not a genre of either kind of metal since it evolved from hardcore punk. But it's always associated with metal, most notably death metal. I guess you could consider it "False Death Metal." I dunno. Whateva
Funk metal; duh!
Industrial Metal; you get the picture.
While true metal genres, with the exception of Heavy Metal for being a fusion of Hard Rock and blues, are all created because of the musics characteristics and themes;
Thrash Metal; fast, aggressive
Power Metal; fast, aggressive, and epic-themed
Black Metal; faster, technical drumming, anti-christian influenced
Death Metal; Technical music, heavy and loud, themes revolve around death, metaphorically or bluntly.
and also, there are crossovers which involve only these true metal genres and not other forms of music, as well as subgenres that strongly emphasized certain base characteristics;
Blackened Death Metal
Symphonic Black Metal
Death Thrash
Melodic Death
Brutal black metal
AGAIN, this is all based on my opinion. There is no set guidelines as to what true metal is, despite what Manowar has lead you to believe.