In the context of the recent forum debates, these are the definitions:
PvP - Player vs player enabled, players can shoot and kill other players and griefing is allowed
PvE - Player vs Environment, players can still play with other players but they can not hurt each other, strictly a "play with other players" environment cooperatively.
Elite's Open is a PvP game mode, while The Mobius private group is intended to be a PvE mode, where commanders play together peacefully, no griefing.