Someone's feeding you bunk...
It's the CARBON CONTENT of the steel and RATE OF COOLING that determines steel's final hardness after heat treatment.
As for quench cooling rates, brine (salt water) cools fastest. This is followed by plain water then oil and finally air. But even cooling in oil happens so quickly there's no time for the steel to absorb any carbon. Rather to absorb significant carbon a part must be maintained at very high temperature (1,500+) for several hours in a carbon-rich environment such as a metal cyanide salt bath, carbon monoxide gas, etc.
Incidentally aside from cost there is no advantage quenching in used motor oil vs new. I suspect used oil will likely smoke and smell worse however.
Regarding cutting oils it is extreme pressure and temperature additives that make an oil effective for cutting metal. Years ago these additives were primarily lard and sulfur - thus the distinctive smell inside every tool and die shop. I have no idea what's used in cutting oils today.