Then WP moves on to the test which Bryan first talked about in a comment to my post and I posited was probably the better test - the use of commas as markers for descriptive phrases (not present in restrictive phrases). He agrees that this is a better test than the judge's offered that/which test. He quoted Strunk and White (and I may have to go out and buy the book because the example is dead on for my purpose), however, since I've yet to travel to my local brick and mortar, I'll quote from my 1896 Higher Lessons in English:
The adjective clause, when not restrictive, is set off by a comma.In plain language I think the difference is this:
The puppy, which is a beagle, is cute.
The puppy which is a beagle is cute.
In the first sentence you are describing the only puppy in the room to someone. In the second sentence you are at a pet store staring through a window at 20 puppies and are describing the specific puppy you are talking about. The first doesn't modify the meaning of the sentence; whether you include ", which is a beagle," or not you are talking about the same dog. The second does modify the meaning; "which is a beagle" defines which - of 20 puppies - you are talking about.
With that, I'm off to my local brick & mortar to purchase an English text published more recently than 1894.