With respect to the maze finder you're mentioning (that guy that put in inputs is me), it's fine since it's something that theoretically can be done without it. To me, a tool typically shouldn't be allowed if it's something you can't theoretically do without it.

The tool used here is effectively a dictionary, where you put in a few inputs and it spews out the pattern. This mapping is not random. Theoretically a human can just memorize every possible input and pattern and just react, the same way as a tool like this. Or, if you wanna look at it this way, it's like a sheet of notes mapping every possible combination to a maze map.

On another hand, something like a "memory reader" should often not be allowed IMO (though this depends on the game). Like, if you take sonic adventure 2 and have a memory reader that tells you all the random piece locations before the level starts, I would constitute that as "cheating" since it's impossible to do that without the tool, and you can't really "write notes" for this.