Jim (PassatDoc) your teacher was wrong.
I do agree that we have some very over-the-top old ladies taking it a bit too far with cleaning, but language-wise, your teacher is plainly wrong. I can only account this to her/him having a certain predjudice thus making the diffrence between "sauber machen" and "reinigen".
Here is my stance (plainly along the lines of language, not about soap consumption): "sauber machen" and "reinigen" are identical.
Of course, once you take it to a more supernatural or esoteric point, then "sauber machen" would not appeal (imagine a vampire movie about "cleansing" your soul, then "reinigen" would be the appropriate term, same thing in literature).
But in normal day-to-day life, both verbs are used in an identical way (hence your teacher is wrong).
Here are ALL the verbs:
"sauber machen" (to make clean = to clean (used in a generic way, no matter what the method is)
"reinigen" (=to clean, to clense, generic as well)
"läutern" (=to clean a soul, make it a better person) - a spiritual term
"desinfizieren" (= to disinfect)
"spülen" = to rinse, also (with china and silverware = to do some washing up / to do the dishes)
"putzen" = to wipe with a wet rag = to sweep floors, to polish things, to clean objects
"waschen" = to wash, to launder (usually meaning using major amouts of water / being different from "putzen" where you only use moderate amounts of water but add more elbow grease to the rubbing action)
"bohnern, blockern" = to polish floors, to wax them and after this, to make them shine ("blockern" is derived from the "Block/er", this heavy metal brush block broom to mechanically buff the floors)
"aufmotzen, aufbrezeln" = to pimp up (your appearance), to use cosmetics and all means of beauty business to get ready to go out (motzen meaning to be infamous, to be fresh or mean or loud-spoken while brezeln meaning to shape in a brezel way = making a gift wrapper out of yourself, think red ribbon around a surprise package here).
"schleifen" = to sand / to scour (using harsh abrasive power to take off material thus gaining a smooth surface)
"scheuern" = to scour (like with scouring powder) or to scrub (with a scotch pad scouring sponge)
"wienern" = to polish to a high shine (boots, chrome, stainless steel) - derived from ""Wiener Kalk" (chalk of Vienna, a very finely ground abrasive polishing powder, now in several polishing pastes such as Amway's chrome polish and similar products)
"wichsen" = to wax, to polish and shine (formerly common, now not really appreciated) = coming from "Schuhwichse" (shoe shine) = to rapidly move along a surface to make it shine - but today only associated with "to wank / j.o." (excuse this, I am only throwing some light on vocabulary).
Please forward this to your teacher. Thx.