On a punching bag, you know you have a snappy punch if your punch strikes the bag, making a visible dent in the material, and that rather than creating a wide swing the bag will violently rock in place with the bottom and top moving in different directions. In addition that, you want to develop a feeling of relaxation in your body, clinching your fist as you hit the bag, and withdrawing your attack as soon as you feel feedback with a relaxation of your grip (among other things), rather than shoving the bag.
Your hand will return to your face as fast as it went out. The feedback from the bag will help with that.
When your punch "pushes" you may initially get the dent and you may initially see the bottom of the bag rise towards you as you hit, but instead of drawing back, relaxed and as fast as possible, you give the bag another 10% causing the top of the bag where you are hitting to be shoved away. The bottom of the bag will follow, and the whole thing will sway in a wide arc. You will have a sensation that your hand came back to your face slowly, or that you felt the bag against your hand for a long period of time.
You can "snap" your punch into the bag by withdrawing the strike after making the dent and feeling the feedback from any range, even with only a few inches to strike, just like you can push something with your arms in tight or your arms extended.
You can almost think of the push as an added extra to the punch, on top of the hit. You have hits, and you have hits + pushes. You can have "pushing only" but you would be aware you are doing that.
Edit: so my main point is that punching mechanic has little to do with "pushing" vs. "snapping." You can push or snap with a 2" punch, and arm punch, a full punch, a major whipping right, or anything else. Snapping is the intention to relax and withdraw instantly upon getting the feedback from the bag after making the dent. Pushing is grinding in and digging (longer, not harder) for that extra 10%. Snapping is mentally easier from a balanced stance, but still doesn't require it. If you are leaning on the bag and snap you will probably fall forward.