None of this are true.
If I remember correctly from what I've learned in Latin American History, there are two or maybe three theories to this.
The first is that the Mayans thought that "time" is cyclical. So presumably, after 2012, begins a new era, or "age", and the cycle repeats.
The second theory is that the Mayans, who carved their calendars in glyphs, you can understand, is in itself a difficult undertaking. When we create our calendar to last our entire lifetime, probably even for our children and grandchildren, at a certain point, you say to yourself, "wait a minute....this calendar will last at least several generations. I'll end at this random date, and if my successors need to create more calendars for themselves, they can do it by themselves."
So it's presumed that the Mayans created their calendar system, that will last them several generations, but they stopped at a random date, in this case, 2012, because they figured...they've done quite enough calendars that should probably last them for a while. Why create more?
And then there's the third theory, which has become very popular, and that is the alleged "doomsday". There currently are not enough credible historical evidence to support this, and not to mention, that the Mayan language is very difficult to decipher. The Mayan language does not have a standardized system....they vary in virtually successive generations, and so it gets really confusing...for example, the letter A will evolve into at least five different shapes/forms within a century...or something like that. And that letter "A" will look starkly different from each other, coming from a foreign observer.
I know of a particular group, the Aztecs, who in fact believed in a "redeemer" date, a date when a person would come, either rescue the Aztecs, or bring about their destruction. This god is Quetzalcoatl.....unfortunately, they were mistaken to assume Hernan Cortez was this supposed "white" God. And the Aztecs and the Mayans have very similar, if not, overlapping cultural connections. But as far as the Mayans go, I doubt very much that they themselves subscribed to the "doomsday" theory....it's more likely that their calendar was either "cyclical", or just an arbitrary end date because they simply got tired of creating glyphs that will last them way beyond their own lifetime.