It may not be correct but I tend to think of a tire as the rubber part, a rim as the metal part and a wheel as the entire assembly .....
Now, as far as some of the other things mentioned in this thread:
I always understood a Monkey Wrench to be like a pipe wrench but without teeth. It is for turning fittings and/or things with heads. A pipe wrench is for turning/holding pipe, therefore it needs teeth to bite into the pipe. A pipe wrench can cause unwanted damage to brass fittings
Now, as to a "left-handed" one ... well, that is entirely another subject