I would define a "punchbowl" as a putting surface where at least three edges slope inward into the putting surface, rather than sitting level or sloping away outward. For the examples you mentioned (referencing my yardage guide to confirm)...
At No. 1, both the surrounding fairway and greens themselves feed inward at the front, left, and back. It sits level with the right. That you disagree with this being a punchbowl is a matter of interpretation. It's certainly not quite as drastic as many Raynorian entries (e.g. Fishers Island) however it also offers more gathering coverage than some Raynors (e.g. Yeamans Hall). Your point that a ball can leave the right side, thus disqualifying it as a punchbowl case, is a legitimate argument. Not one I agree with, but a legitimate one!
At No. 2, looks can be deceiving because it appears like a false front, due to the fairway fall-off at the front. However as soon as one reaches the front of the putting surface, the green itself flows down toward its center, and this is true of the back and right sides as well. I'll note that I am beyond satisfied with this instance.
Using my own definition, No. 9 is disqualified because it only slopes in from the left and right sides. But with a green that long, that's a vast majority of the edge coverage. Somewhat like Potter Stewart's take on pornography, I will argue that "I know punchbowl when I see it."
As a counter to give you an example that I am NOT referencing, the feeder slope to the right of the No. 4 green is NOT an example. In that instance, the bold player may fly directly at the stick, risking the chance of landing in the waste left, in part because there is no inward slope on the left side to give you an airbag bounce. A more risk-averse player can try his hand at guiding the ball down via the feeder slope, however...less guaranteed accuracy but less risk as well. On No. 11, however, if you choose to take a direct line over that deep pit ahead of the green, the putting surface slopes inward to assure that if you get it there, it will move forward, not back. The latter is not a "punchbowl" green in the least, however it uses punchbowl techniques as insurance. Perhaps some would prefer to call them "feeder" techniques.
How do you define a punchbowl green, Ryan?
I just went through the holes and I don't know that I see as many punchbowls as you do. There are slopes that will roll the ball onto the green, but to me, a punchbowl has four, three, or at least two sides that will do this (two can get iffy, IMO, as it could just be a back and a right side or something, with a false front and left… and that probably wouldn't be a "punchbowl" to me).
For example, do you see the first as a punchbowl? I don't. You can bounce or roll the ball on from short left, but the ball will funnel off the back right pretty easily, too, as the whole topography around that green is in that direction (front left to back right). That isn't a punchbowl to me.
No, it's not raised greens like at Pine Needles or #2, but I don't know that I agree that TR is full of punchbowl greens.