Yes, rep is short for repetition, and yes rep is the same as repetition in the given example. However, the definition of repetition in the example is not the same as the "usual" one (the one I think you're thinking of). Consider the following definitions:

repetition

1. a : the act or an instance of repeating or being repeated
   b : a motion or exercise (such as a push-up) that is repeated and usually counted

(M-W)

In the sense of 1a, yes, the first one is not a repetition, but the following nine pushups are. However, in weightlifting, rep and repetition often mean 1b. In the sense of 1b, the first pushup is also a rep or repetition even though it's not repeating a previous instance. So if John does ten reps, he does ten pushups. Similarly, if he does ten pushups, he does ten reps, not nine.

Here are two more excerpts:

• rep
  (in bodybuilding) a repetition of a set of exercises.
  ‘complete all reps on one leg and then change over’
  (ODO)
• Weight Training: How Many Reps (and Sets) to Do
  You can’t design a strength-training (or weight-training) program without knowing two terms: rep and set. Rep (repetition) is one complete motion of an exercise. A set is a group of consecutive repetitions. For example, you can say, “I did two sets of ten reps on the chest press.” This means that you did ten consecutive chest presses, rested, and then did another ten chest presses.
  (Dummies.com)

This usage is definitely idiomatic in the realm of weightlifting, but it might not be so natural in other circumstances.