There are two definitions of alpha male that are at odds with each other.
I brought it up in a thread a while back, but it got no replies. Maybe the wording was bad.
I've noticed different spheres in the manosphere define and rank beta males completely differently. For those coming from a more PUA side (most of this forum), there are only alpha and beta males. The beta male is seen negatively as a weak chump who gets settled for and pays the bills in a...
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Alpha male in manosphere terms is usually "big rich muscle guy who has lots of sex, can yell loudly, and wants everyone to know." This is what gets mocked. Young men puff up their chests and try to mold themselves into an 80's action movie hero character primarily to attract women (and it's very obvious how strange it is). This actually does result in more sex and being more attractive, though.
Alpha male in reactionary terms is more of a commander of loyal beta males or one who resolves conflicts between parties. Women don't have anything to do with this definition, and they might not even be attracted to this type of man. This is the alpha male most people immediately recognize and respect, even if it's a reserved, nerdy guy like Garry Kasparov. Kasparov manages his own entourage, and those men wouldn't be loyal to him if he offered nothing. Beta males here are more like a CFO to a CEO, in business terms.