What's on UO.com (what you linked to) is basically the Faction backstory.
Minax invaded, and basically won, "taking over" Britannia. Nystul created Trammel, so we could escape to it. Thus, there is an RP explanation for the Trammel/Felucca split.
Dupre was supposed to lead the effort to reclaim Felucca: That's the True Brit. Faction.
The Council of Mages blamed Nystul's magic for the devastation of Felucca (Nystul blamed Minax) and objected to it. They also objected to a secret meeting between Nystul and the Time Lord. So, they splintered off, switching to Magincia (in Felucca anyway). And that's the Council of Mages faction.
The Shadowlords Faction was created, for evil and mysterious reasons, by a fellow named Ramuz. (Years later, he was fought by players as part of the post-AoS EM events, and the last we heard he was in the Lord British's castle dungeon, which we never get to see.)
In some cases, you can plug any of these names into UOGuide.com for some more information.
-Galen's player