Well there are many variants of teams and it is hard to list them all because making a new kind of team can be done without you noticing it. You have mentioned the most popular but I will just add on a few:
Rain teams are often lead by Politoed in the OU tier and Kyogre in the Uber tier. They summon permanent rain. Rain teams often consist of Pokemon that can abuse Thunder/Hurricane and that will usually have a fire weakness. In rain the fire damage is reduced by 50% and the water type moves are boosted by 50%
Sun teams are often lead by Ninetales in the OU tier and Groudon in the Uber tier. Theysummon permanent sun which will reduce the power of water type moves by 50% and boost fire type moves by 50%. MOves such as Thunder and Hurricane have a 25% of accuracy
Sand teams are often lead by Tyranitar and or Hippowdon. They both summon a permanent sandstream that will activate certain abilities that will power up ground type moves are boost Special Defense. All non-steel, rock, ground Pokemon will be buffeted by the sand stream each turn
Hail teams are often lead by Abomasnow that summons a permanent hailstorm. Moves such as Blizzard will have a 100% chance of hitting. All non-Ice Pokemon will be buffeted by the hail each turn
Trick Room teams usually consist of very slow Pokemon and often 2 Pokemon that twist the dimensions
Stall Teams usually consist of bulky Pokemon that have more non-damaging moves then damaging moves. With moves such as toxic, rest, recover etc.. they stall their way to victory.
Monotype teams consist out of 6 Pokemon that share one type
Hyper Offensive teams
Hyper Offensive teams consist out of 5 Pokemon that have only attacking moves and that have a high offensive presence and 1 Pokemon that lays entry hazards. The attacking Pokemon are usually frail
Standard Offensive teams
Standard offensive teams are very similar to Hyper Offensive teams but rely much more on speed then the 2 other forms of offense. They often utilize powerful choice items and scouting moves such as U-Turn.Good prediction is what keeps standard offensive teams going
Bulky Offensive teams
Bulky Offensive teams rely on bulk rather then speed. They are designed to take hits well and react with an immediate punch in the face
These teams are made of Pokemon which were all released in the same generation - however their Moves and Abilities do not need to be of the same generation
They're all of the same main colour. Yay.
These teams focus on using the move Gravity, which allows high powered, low accuracy moves like Stone Edge and Focus Blast to hit alot more reliably.
So they go on. Usually subs or combinations of 2 kinds of teams come under these teams such as Sun Hyper Offense or Rain Bulky Offense or Sand Stall.