The cactus in my yard don't even need to be planted. They lie on bare dirt or turf and send out roots on their own and send out new upright pads from there. (These are prickly pear and other pad-style cacti).

The columnar cacti may or may not respond. I would say "keep all of the parts" and go from there. Stop watering the bottom, but cut it off at the point that looks constricted and lay the top portion on a pot of well-drained (lots of sand in it) dirt. Don't over water or over fertilize either of them. The one still rooted might heal over and send out a new segment if it still has roots. The beheaded one might send out roots. Might. Cactus are tough, but they don't respond well to TLC, so resist poking too much.

SRS