It's like trying to define the differences between artistic and obscene pornography... I know it when I see it, it's all subjective and you may not agree with me.
You may take a class and think it was training to fight, I may come away from the same class thinking it was the equivalent of NRA 1st steps.
I think the students background and experiences have more to do with it than the instructors.
I know instructors who claim to teach "fighting with a gun" and never come close, the other side of the coin is I know instructors who advertise a pure skills course and devote significant time to "fighting with a gun".
Then you have students who take things at face value and sometimes miss the underlying lesson in some drills.
There are definitely classes that emphasize the martial aspects of shooting and there are classes that emphasize the mechanics.
Would you get the same material from Southnarc and Brian Enos, or would one better prepare you for a street fight and the other prep you for an IPSC match?