Alpha and beta behavioral traits and alpha-beta-gamma-delta-sigma-omega-lambda status in sociosexual hierarchy are different.
The Alpha Male of the group(AMOG) is the leader, the center of attention, the classic star of the football team who is dating the prettiest cheerleader, the successful business executive with the beautiful, stylish, blonde, size zero wife. On the other hand it's a hugely stressful lifestyle, as status is dynamic and if you aren't at the top, you'll lose those extra benefits you gain from that status very fast.
Beta status isn't inherently nice, you can be the beta in a gang. It's actually not a bad spot, just one step down from the highest, not suffering any leadership challenges and still benefiting a lot(though not as much as an alpha) from the total status of the group.
Beta behavioral trait however is vastly different. It is the traits that is needed in the modern age. It is the traits that build "comfort" or "rapport" in a relationship.(As described by SimplyAmorous in the previous post.)
Edit: On the other hand though, although it's a lot of fun, such complicated methodology isn't really that necessary.
Edit 2: Different bloggers have different ideas on such hierarchy. For example Roissy's hierarchy goes alpha(good)-beta(bad)-omega(worse). Athol's goes like alpha(not enough beta)-beta(not enough alpha)-gamma(balanced alpha and beta traits)-omega(devoid of such traits, date this man at your own peril.).
The one to pick really depends on what your endgame is, actually.