The Crimson Room and the (sort of a) sequel Viridian Room by Toshimitsu Takagi were very popular when they came out, although Viridian Room is better in my personal opinion. Unfortunately, both do require flash player and aren’t on itch, but easy to find with a quick duckduckgo search.
When someone asked me if they were puzzle games and would only accept “Yes” or “No” for an answer, I would go with Yes: Like a puzzle game has more than just one level, an escape room game has more than one type of puzzle - they are just connected by a more intricate level select screen that is also sort of a meta-puzzle where every other puzzle is one of the puzzle pieces. So if anything, an escape room game is more of a puzzle game that your regular puzzle game - but more diversified.
I hope this helps.