I tend to disagree. Air Coryell, West Coast, Spread, Run'n'Shoot, Smash Mouth those are all philosophies. A philosophy provides the framework how things should get done, e.g., the Smash Mouth heavily favors the run game to dominate time of possession and to be able to successfully use play action passes, whereas the West Coast offense favors short quick passes instead of a heavy run game. That's your philosophy.
A system is the detailed description of how to do things within the philosophy. For example, you can have a West Coast philosophy, but you can have those short quick passes out of the Shotgun, out of the Singleback, out of the Pistol, out of a thousand formations. Also, the terminology will be different from coach to coach. That's what defines your system.
That's how I see it at least.