There is no rest between exercises - only after sets, unless specified otherwise. You have to complete the entire set going from one exercise to the next as fast as you can before you can rest. What does “up to 2 minutes rest” mean: it means you can rest for up to 2 minutes but the sooner you can go again the better. Eventually your recovery time will improve naturally, you won’t need all two minutes to recover - and that will also be an indication of your improving fitness.
Recommended rest time:
Level I: 2 minutes or less
Level II: 60 seconds or less
Level III: 30 seconds or less
If you can’t do all out push-ups yet on Level I it is perfectly acceptable to do knee push-ups instead. The modification works the same muscles as a full push-up but lowers the load significantly helping you build up on it first. It is also ok to switch to knee push-ups at any point if you can no longer do full push-ups in the following sets.