Proof is actually a function of intention - special preparation of the dies and planchets. It may also be struck on special presses, double struck, etc.
Cameo is a function of the actual strike, highly depending on the instantaneous state of the dies, e.g. polished and then just enough strikes to wear away the scratches leaving a mirrored surface.
Many cameo coins are proofs, but there are also cameos found (rarely) in business strikes.
The other terms (e.g. deep cameo) are modifiers that indicate how reflective the surface is. For example at one grading service, you must be able to read newsprint at 10" (or something like that).
-----Burton
50+ year / Life / Emeritus
ANA member (joined 12/1/1973)
Life member: Numismatics International, CONECA
Member: TNA, FtWCC, NETCC, EveryCountry (online) coin club
Owned by three cats and a wife of 40+ years (joined 1983)
Author: 3rd Edition of the Sample Slabs book,
https://www.sampleslabs.info/