professor Could be wrong, but I don't think the video footage is definitive.
At the courses I play, the putting surface is clearly delineated from the fringes because the grass on the putting surface is actually green while sometimes the surrounding areas are, let's say, not quite as green.
That's an expensive course, and the green appears to have a false front that is closely mown and extremely green; from that angle it's hard to differentiate that part of the fringe from the actual putting surface (and if you think about it, that's kind of the point of a "false front." If you could clearly tell it wasn't part of the green, it wouldn't serve its purpose).
Not backing up Trump's actions here, as I said, because the cart should even be NEAR the putting surface. I'm just saying that it's not completely clear that his cart is actually on the putting surface.
(If Trump wants to drive on the greens he should get a hovercraft golf cart like Bubba did. Does no damage to the greens at all. Plus it's more fun and you can also drive it right over the water hazards.)