From the FF13 datalogs:

"The Guardian Corps is the branch of the Sanctum military responsible for maintaining peace and stability within Cocoon.
Within their assigned jurisdictions, the soldiers of the Guardian Corps fulfill the role of police officers and protectors, working to eliminate all manner of criminal activity as well as rampaging wildlife."

Cocoon only has PSICOM and Guardian Corps, hence Guardian Corps is the name for their regular military.

Yes the level of training was higher than Cloud, I think I have said that in every single post I have made. But it still wasn't training anywhere close to the level both of them are currently performing at.