Alpha and Beta do not exist for our species. With the way people use the words they don't actually exist in any species.
People use Beta when talking about someone who is passive and doesn't take charge, when in the animal kingdom Beta animals are second in command. They lead when the Alpha pair are not around, and they become the Alpha pair if something happens to the original leaders. There's nothing passive or subservient about them, the word for that caste is Omega.
Some also tie it to the A and B personality types. Besides the fact that it's almost a 70 year old system and has been overshadowed by many more accurate ways of measuring personality, it also doesn't perfectly match the way people use Alpha and Beta. It's basically a system that describes Extroversion and Introversion, with a few extra details attached here and there. Contrary to popular belief Introverts can be great leaders and very outgoing people.
Then you get people who, and though I don't have the link I'll try to quote the guy as best I can, "get all the pussy and everyone else is a Beta cuck". I literally had someone say this to me on the boards, claiming to be a "real Alpha male". These people wouldn't know basic psychology if it hit them with a car, and are honestly infuriating to share a species with. They are hyperdefensive and show just about all signs of having a superiority complex. I'll see if I can find the thread, though I doubt it with the quality of the board's search function.
Now, what the OP is talking about is taking responsibility for your own actions, not being a blame magnet. If your goal is to win you should try and help your allies as best you can, and recognizing what you could have done differently allows you to improve your decision making as you get better at the game. You don't have to take the blame, you don't have to say "sorry guys, it's my fault for not doing X", you just have to realize that doing X could have tipped the game in your favor, and make a note to try and do better next time.
It's about self criticism, which is a very powerful skill many people never developed. Being able to blame yourself instead of others is just an important first step.