An Alpha is a leader above all else....
Being "Alpha" all boils down to being a leader. Now you have different types of leaders, all with different characteristics, which you effectively pointed out in your post.
I'd just like to add that so many guys focus on copying or mimicking the mannerisms of a dominate male (A.K.A. Alpha Male). While there is nothing wrong with this, I think men would be far more effective if they let their own individual qualities emerge, rather than simply copying the behavior of another. In other words, just be yourself (I know some may cringe after reading that last sentence, but that's what game ultimately comes down to). If you can confidently and assertively express yourself without apology, you'll be "alpha," and there are different ways to be alpha, as ilikecharlene wrote.