There are three types of Beta mentioned on this site. These are:
Testing Beta Edit
A testing Beta is what Jagex use to create games. Much like the RuneScape Testing Beta they usually have features added to them and then are swapped with the live game.
Early Version Taster Beta Edit
An early version testing beta is a game that is just released before the game to test bugs and see how well the gameplay works as well as to give a taster of what. There have only been two betas like this and these are Kickabout league and Runescape HD. These games were both released as betas before being fully released.
2nd version Beta Edit
A 2nd version of a game can also be a Beta. E.g. if a game is re-made instantly (not if its re-made after a few years since the last version) or slowly changes so that the whole game has changed then the finished game would be the Beta.