So just to kick the hornets nest here...if you are hitting 135 at Hudson, you are absolutely under overhead...and given that there is zero natural light down there, technically there is no cavern zone....
That said, I would have no problem using Hudson for deep training, other than the fact that it makes your gear smell like rotten eggs.
Umm, I guess if you want to split hairs, you may be right. But let's be realistic here, the horizontal penetration doesn't extend beyond 10', and the shape of the basin is more like the walls of a bottle, following the only wall will bring you straight to the surface.
The map is here:
The only course I use the 130' section for is AN/DP -- a course with a physiological overhead already, the pre-reqs to enter are over 75 dives and some technical training, and fully redundant gear.
Now let's get back on the topic of using Blue Grotto for the deep dive in Advanced, or the cavern for OW training. Students in their OW course are, well, not even certified divers (yet). And the pre-reqs for Advanced are typically just being open water certified, so it would likely that the deep dive in Blue Grotto would be their 6th or 7th dive ever. And what about their gear requirements? Basic open-water kit.
Now, let's look at the map of Blue Grotto.
It looks like you have to have approximately 100' horizontal penetration to get to the deep section of Blue Grotto. And you have to go around several obstacles (boulders, rocks, restrictions) to get there.
How is this even similar?
Let's go back to Hudson for a sec (map above). There's a down line and platform at 100' right in the center of the sink. Instructors can easily use that platform to accomplish the deep dive in an Advanced class without ever getting near an overhead, but may prefer to use Blue Grotto.