The only colloquial usage I've seen is that it is someone who has the ability to demand that others perform a public display of obedience to them. Like every time I've seen somebody unironically refer to themself as an "alpha," it's always had that underlying connotation of "Respect Me!" And every time I've seen someone mocking somebody else for being an "alpha," it's because, well, that respect was clearly undeserved.