Also, in that episode ["Chain of Command"] the Enterprise was also called the flagship of the fleet.
Nope, Nechayev again uses the term "Federation flagship" - no mention of "fleet" or "Starfleet" there, and thus no contradiction with the usual naval meaning of flagship.
Rather, full compatibility with Evans' take on the nomenclature: the E-D is simply the showpiece vessel of the Federation, and it's quite incidental that she also happens to be serving in the Federation Starfleet and performing a command task every now and then.
The ship is also called "the command ship for the sector" by Gul Madred. This is a plausible military designation allowing for a CO of Captain rank and not interfering with the concept of "flagship" in any way.
Timo Saloniemi