That’s a really great point.
Since this is a forum for Japanese-as-second-language instruction, the third school you’ve listed gets my vote! 
In all seriousness, the comments in this thread essentially suggesting that it’s better to “think in Japanese” to analyze Japanese grammar really resonate with me.
I’ve occasionally had an incredibly difficult time when someone explained a Japanese sentence using Western grammatical jargon. It felt like looking through at least a two layers of dirty windows: my knowledge of Western grammar terms is shallow at best, and Western ideas usually don’t map 1:1 into Japanese without some rather tortured logic.
To be fair, using Japanese 文法 terms would be just as opaque to me.
The grammar explanations I have the easiest time grasping come from Jay Rubin’s book “Making Sense of Japanese” and the Cure Dolly videos. I previously thought it was because they were aimed at beginners, but after you made this point I realize it’s probably because they are by and for learners of Japanese as a second language, and they expressly warn against trying to apply Western ideas too narrowly.
Well, my inadequacies are now feeling pretty exposed. Anyone else?
I’d like to formally invite you and @Kazzeon to the sentence diagramming thread. The above would be MUCH easier to parse with some sort of diagrams showing the breakdown! We are trying to figure out a simple visual language to at least show the syntactical breakdown of sentences like this (without going too far into semantic meaning).