I think the distinction is between what is known as the "Olympic press" vs "strict press" and clean and press refers to the competition Olympic press. When originally included in weightlifting, the press was very strict. In fact, the head referee would elevate his hand and the lifter would raise the bar accordingly.
Over time, the press degenerated into a quick lift as lumbar extension (layback) with bent knees was allowed to start the lift, and the torso layback (a second layback) at the end as it is easier to lower your body under the bar than raise it. The judging became increasingly political, and certain lifters could get away with more or less knee kick and layback. The head referee gave a start signal and sometimes as the lifter would lean back to start the knee kick, there would be no signal given until they straightened up.
It really wasn't cheating if the judges allowed it. However, this eventually led to the elimination of the press from competition in 1972. Google Bill March, arguably one of the best pressers ever. He made an incredibly strict press with 390, at 198 bdwt., at the 1969 Senior Nationals. No leg kick or layback. I believe there are pictures out there somewhere.