I see your point about The Witness. It has been a while since I played it, but now that you mentioned it, I agree it contain puzzles of both types. Better examples: A Good Snowman Is Hard To Build, Stephen's Sausage Roll, Trainyard, Puzzle of Jellies.
Regarding your suggestion about the search, I guess it is helpful, but my point was that they are just no belong under the same category. I believe the appeal to different audiences.