Here are my guesses on factions:
1. Those Loyal to Joffrey Baratheon (the incestuous child of the Queen (Lannister) and her brother Jamie)
Joffrey (presumably, though he could be with the Queen, see below)
Tyrion (Savior of the City)
Sandor Clegane (Kingsguard)
Ser Balon Swann (Kingsguard)
Ser Jacelyn Bywater (some kind of commander of the city guard)
Littlefinger (went to recruit Tyrells)
Tywin Lannister (the patriarch of Lannister, lead arriving force just in time to save the day, right flank)
Mace Tyrell (lead the arriving force, left)
Randyll Tarly (lead the arriving force, center)
Ser Garlan Tyrell (vanguard of the arriving force - wore Renly's armor creating the myth of "Renly's shade")
This group is probably town, being the defenders and the most numerous.
2. Those loyal to Stannis Baratheon (the bastard child of King Robert, but blood related to him unlike the Queen's children with her (twin) brother.
Stannis
Ser Guyard Morrigen (vanguard)
Ser Rolland Storm (conducted rearguard during retreat, allowing ~2000k warriors and Stannis to escape)
Ser Imry Florent (commander of the invading fleet, proceeded recklessly into the trap)
This group could be Mafia or a third party faction)
3. Those loyal to the Queen (we'll call them Clittorians, as there is no official designation for the Queen and their cronies)
The Queen
Ser Mandon Moore (Tyrion's sworn shield, attempted to kill him when Tyrion lead a sortie against the boats that managed to make landfall)
Grand Maester Pycelle (doesn't appear at the battle, but was giving information from Tyrion to the Queen via the small council)
This group could be the Mafia or a third party faction.
4. "Others" (the creatures beyond The Wall)
They are rumored to be creatures that cause victims to become undead (zombies/wights)
This could be a solo player, an SK or a group and thus the Mafia.
For other names of those involved, see this link (from the OP), especially "Honors and Rewards"