what's the difference between a fake alpha male and someone but didn't use to be an alpha male but has since become one?

Congruency. If someone became an alpha male by PUA training, they will probably have the "visible" traits of alpha male, but lack the "invisible" traits (where "invisible" is a short for "not easy to detect during the first date"), because the training will focus on the "visible" traits.

Unless it is a PUA training that explicitly focuses on teaching the "invisible" traits (because they believe that this is the best way to learn and maintain the "visible" traits in long term).

At this point people usually begin to discuss the definition of the term "PUA". People who like PUA will insist that such trainings belong under the PUA label, and perhaps they are the ultimate PUA trainings, the results of decades of field research. People who dislike the PUA will insist that the label "PUA" should be used only for those surface trainings that create "fake alpha males", which is a bad thing, and that any complex personality improvement program is just old-fashioned "manning up", which is a good thing, and should not be confused with the bad thing. This battle for the correct definition is simply the battle for attaching the bad or good label to the whole concept of PUA.

Good story. Someone who didn't use to be an alpha male but has become one often has a good story that explains why it happened. A story "first I was pathetic, but then I paid a lot of money to people who taught me to be less pathetic, so I could get laid" is not a good story. A good story involves your favorite dog dying, and you being lost in a jungle after the helicopter crash, feeding for weeks on scorpions and venomous snakes. Or spending a few years in army. If a miracle transformed you to an alpha male, your past is forgiven, because there is a clean line between the old you and the new you. Also if the shock was enough to wake you up, then you probably had a good potential, you just didn't use it fully; you had to be pushed forward, but you found the way instinctively.

There is a fear that if someone gained a trait too easily, they can also lose it easily. (Imagine the shame of being known as Joe's former girlfriend, if Joe returns to his previous pathetic behavior, because the PUA lessons did not stick.) And if their gaining the trait was based on education, not genetics, then what is the point of getting their genes? :D

Is someone who didn't grow up speaking English but now does a “fake English speaker”?

A proper analogy would be someone who memorizes a list of English phrases frequently used in some context, with a perfect accent, and then meets you in that context to make a good impression. Only when you ask something unexpected, it turns out the person does not understand most English words.

Of course there is a continuum between fake English knowledge and real English knowledge, but people are expected to cross the continuum in a predictable manner (gradually getting better in all topics). When people speak about "natural" and "fake", they often mean "predictable and reliable" and "optimized for cheap first impression". If someone knows 20% of English words in any context, then 50%, then 80%, then 98%, that is learning; if someone knows 100% in one context while knowing 5% in other contexts, that is cheating -- this path might finally take you to the same goal, but the mere fact that someone is using this path suggests that they are too lazy to finish it.

Don't lots of men drink alcohol in order for women to look more attractive to them? :-)

I am not sure if this describes the real behavior, but supposing that they do, they do it voluntarily.