Well, you pretty much hit most of it in two terms...

Know ALL aspects of your kitchen (including procurment of supples, food costs, portioning, menu development, food safety management, etc. and managing staff, etc.)

and...

Know food (Including history, all preparation methods, not just the how, but the why, also baking, even if you employ a baker/pastry chef)

An experienced chef is also wise to front-of-house/dining issues.

A trained "cook" rarely makes the claim of being a chef, especially if they are within ear-shot of a true chef, UNLESS they have been responsible for an entire kitchen operation, from menu planning to close-out. Lead line cook doesn't qualify, no matter how many friends call you a chef. And if you aspire to be a chef... get an early start. You are going to need the stamina of a young buck! And legs of steel.