Many people believe proof coins must be uncirculated and some even believe a coin that is circulated can’t be a proof. A few others believe a business strike (pocket change) coin that is uncirculated must be a proof, but that is not the facts. I know all this sounds circular but this is the three incorrect assumptions on proof coins I have heard people claim.
A proof coin is struck with dies that are specially prepared, many have some slight design differences to help identify them as proof and the proof die pair strikes the coin more than once to gain a sharp and detailed strike.
It is also not the same as proof-like or deep mirror proof-like since these are coins that were intended for spending but were either early die states or dies that were recently polished to remove clash marks.
An example is a proof-like Morgan Dollar; they’re not proof coins and do not have the same strike or die marks as an actual proof Morgan Dollar. Almost all proof coins have die marks and characteristics that differentiates them from a business strike coin.
Below is a 1952 Franklin Half Dollar Proof that was lightly circulated and was graded PF55 by ANACS.