No one should argue with any of this. Thus, I believe that the terms little stud and real stud are appropriate. However, let me state that if your game is little stud and you do well at it, you have nothing to be ashamed of. Let's take a look. First, little and real stud differ in the general thrust of correct strategy.
Little stud is essentially a trapping game, while big stud is usually a game of knocking people out.
Of course there are some exceptions, but in little stud, since you start off with very little money in the pot, it's not necessary to steal the antes as often, and you are frequently trying to entice the weaker players into the pot. For example, suppose your starting hand is a pair of kings with a king up. In little stud you should frequently limp in for the bring-in or make less than a maximum bet in a small spread limit game. The idea is to trap a weak player, and then hope that by fourth street he has enough of a hand to stay with you. Of course you may get beat, but being able to take a big pair up against a smaller pair is a big advantage in this game.