Cheap means "low in price". Price can't be "low in price", it can be just "low".

Cheap has many more meanings than that. It can mean comparatively inexpensive (a cheap Lamborghini), of inferior quality (cheap paperclip), miserly (he's too cheap to buy better), gained with little effort (cheap win).

So what's a cheap price? A price of inferior quality? A miserly price? A price obtained with little effort? Or is after all a low price what's meant?

Sure. Still doesn't make "cheap price" a valid combo.

could be a contraction of cheap at the price https://en.m.wiktionary.org/wiki/cheap_at_the_price