A punch shot with a Push Basic type linear Right Arm extension is Hitting but only one form of it. One very basic, extreme form along the spectrum that divides swinging and hitting.
Sticking to just the Push vs Pull thing and the Right Arm motion , the right elbow is pushing until it crosses over to in front of the Hands and is thereby pulling. So yes the linear shot putting type arm extension is a push but so is a side armed , stone skipping like throw for instance.
There's a lot of different shades of grey in the spectrum between black and white , pulling and pushing. You could conceivably get smack dab in the middle. Or you could Pull then Push instead of just Pushing the whole while.
A very wise man once said to me.... " ALL the guys on tour are trying to figure out how to get their right arms into it".
I used to think Hitters were a rare breed , now I see them everywhere. Most young juniors , women , some top ranked amateurs ...... seniors, especially. But yeah, some guys are hard to define, thats for sure.