What is the difference between a "compass swing" and "check swing" procedure?
Based upon the terminology I have heard over the years, a compass swing is the alignment of the compass, normally using a compass rose painted on a surface. The compass is adjusted and a deviation card is created for cardinal points on the compass.

A check swing, as I have heard it referred to, is the checking of the compass, and done with the aircraft running, and electrical loads on, similar to a compass swing, but without the adjustment.  The deviation card is verified, but there is no intent of adjusting anything.  This was typically done before we ferried aircraft over the ocean, where there would be extended dead-reckoning (before GPS and LORAN), and where flights were close to the polar regions, and compass operation needed to be verified prior to the flight.

Not that a compass is worth much near the poles (grin).

My guess is that check swing is merely a slang term, whereas compass swing is well understood and extensively documented.