AoM alpha was for multiplayer testing a few months prior to AoM full release. The alpha was available to 10,000 people who signed up for it. The alpha stopped working when the alpha period expired.
AoM beta is something that wasnt' done publicly to my knowledge.
AoM also had another promotional prerelease that gave a large number of people the game for free based on a signup which contained 1/2 of the games scenario. This was released a month prior to the games full release.
AoM full release was done on Oct 31 2002 which is what you should get if you wish to play it online on ESO and is the only version for which the deserving developers get any money.
If your not sure you will like the game the AoM demo is available free to everybody and can be downloaded from Ensemble Studios website. There should be no reason to go and get the warez (illegal) version of the game.
The warez (illegal) version(s) of the game wont work on the ESO and you wont be able to play with people who have the legal version via direct IP either. Different warez versions wont be compatible with each other either.
Save yourself the trouble and do the right thing. Get the game by purchasing it. Download the demo if you want to know more before buying.