You seem to imply that all rules are equally as important as one another, while I would say they are not. For example, in a casual game of chess, most would be okay with some of the aforementioned deviations, like taking back a move. I do not think most would accept it if you started moving your pawns around like queens.
For example, under the rules, not moving a piece you touched and promoting to the other color are the same: illegal.
The examples you are giving are simply cases where the other player is likely to let you break the rules if it's a very casual game.
So again, having "strictly" in the sentence is, technically, purely redundant.
It's a dumb thing to say, but then again, it probably made the piece more suspenseful and thus more fun to read.