Maybe the medieval culture actually has a fanatical devotion to magic, and shuns technology on principle. That would explain why there's a massive technology gap.
As for steampunk: Probably exactly as you said: cultural preference more than anything. Probably steampunk tech can do most of the same things as cyberpyunk, just in a much different way.
Empire names:
Steampunk:
Mistvale(an allusion to the ever present steam clouds in the air)
Avolire
Magical:
Avalon
Aurelia
Cyberpunk:
Ecrion
Zoze
Just some random possibilities.
What does Mercenary Inc do, and why are they allowed to do it?