What is "operational" means, that you can design a feedback (negative/positive) in a variety ways, to create any math "operation" - differential, integral or proportional.
That definition is valid as long as you get other people to accept it.
The higher gain, the easier it could be done.
google "open loop gain" and "closed loop gain".
Device with low gain is not "operational". Just amplifier.
Every opamp has gain < 1.0x, and by your definition, they shouldn't be called opamp.
Take some good EE101 will help.