1st playthrough
Inquisitor: woman elven warrior who's all about elven freedom
LI: Solas
Team Humans Are Crazy: Solas, Sera and Varric
2nd playthrough
Inquisitor: woman human mage who's all about mage freedom
LI: Cullen
Team Mutually Respectful Moral and Theological Debate: Cole, Cassandra and Solas
3rd playthrough
Inquisitor: woman human rogue who's all about getting out of that **** alive, thank you very much
LI: Blackwall
Team Grim Determination: Blackwall, Vivienne and Varric
4th playthrough
Inquisitor: male elven warrior or rogue
LI: Dorian
Team YOLO: Dorian, Iron Bull and Sera
5th playthrough
Inquisitor: male human mage
LI: Cassandra
Team I Swear I Will Pull this Inquisition Over Right Now if You Don’t Stop: Cassandra, Sera and Cole
6th playthrough
Inquisitor: female or male human warrior
LI: Josephine
Team Talking, But Not To Each Other: Sera, Vivienne and Cole
7th playthrough
Inquisitor: female dwarven warrior
LI: Sera
Team Crushed Expectations: Solas, Sera and Cole
8th playthrough
Inquisitor: Female Qunari mage
LI: Iron Bull
Team Vulgar: Iron Bull, Sera and Blackwall
Yes, I really want to know why the devs used those names 
It's awfully lacking Dorian though... I'll probably give up quickly and just keep Dorian in my group all the time.