Definitely semantics are an issue here.
I heard these definitions that I think are pretty good: For push putting - if you crouch down some, like bowing, and then throw the putter using the shoulder as a hinge, and your arm goes straight forward towards the basket, that's "push putting".
The reason for some confusion is that there *IS* spin on the disc; you are not throwing Phil Niekro knuckleballs here. The wrist DOES participate in hurling the disc towards the target; the amount of that is up to the individual player. Most disc golfers are push-putters of some kind.
Spin putting is where the arm is usually held up and will come forward at the elbow, and the wrist often (but not necessarily always) will be "snapped" to add energy to the putt. Simon Lizotte has become very good with this putt.
It was either Ulibarri or Feldberg who has this very interesting putting stroke: he brings his arm up and throws an anyhyzer putt to the basket. That's a spin putt, btw. Because I'm so tall, I tried that putting method at baskets, and I still bring my hand up to my chest more on downhill putts, but I found my results inconsistent (maybe if I practiced as much as the pros...)
So what I ended up with was crouching down, and push-putting the disc from my belly. I don't try to hold my wrist rigid, though... I definitely use it to throw the disc at the basket. But I don't use much elbow in my putts; using the shoulder as the hinge brought me a more consistent putting stroke.
Dave Dunipace put out something about a "snake strike", where one thrusts one's arm at the basket like a snake striking at it; this is a good thing to try, and it helped me a lot. I don't do exactly that, but the concept was helpful and I learned from it.
Bottom line: try different things, see what works best, then practice from 20 ft. (give or take). You'll soon know if what you're doing is best FOR YOU, by the sound of disc-striking-chains (or lack thereof). And watch a LOT of videos... what McBeth does is not what Lizotte does is not what Feldberg does, etc. etc. etc. Take the best of everything that works for YOU, and go throw putts until the cows come home....