my area of focus is just in programming, to say i have much expertise in anything would be misrepresentation at best :P.
To break down why I called the first video what i did, I'll go over my reasoning.
I didn't see much adventure. To me that would mean a level with things to explore and do, whereas in this gameplay, leaving the rail and hitting a rock results in a restart.
I call it linear because the level follows one defined path with no exploartion and has a clear start and finish.
My use of arcade refers mainly to the time keeping and score, much like an arcade game.
The use of platformer comes from the fact that it uses common platformer-esque mechanics, like the grind rail, which you can follow, and jump off of, but if you fall, you fail, similar to other more traditional platformer games.
However it is clear that your game will eventually expand to encompass more features, and has things like puzzles and quests, which would fall under your initially described genre of puzzle adventure. I wasn't aware that it was more of a tech demo of a specific mechanic, until I got the missing context. I am by no means a genre expert, just played a lot of different styles of games
I can definitely see all of these mechanics being able to come together into a really fun game :D