The Cactus on the right with the smaller, rounder leaves is the one that is from the old plant and blooms at Christmas time.
These cactus do best left outside in partial shade all spring and summer. Then I move them to cover on the open porch and drape a plastic drop cloth over them at night when temps drop to the 30's. If you bring them in the house, move them while it is still warm so the temp change will not make them drop their buds. They need 12 hours of darkness to bloom so I put them in an unused room or guest bdrm or even in the guest bathtub/shower and bring them out when in full bloom. The blossoms then last for many weeks giving you many days of beauty throughout the seasons!
Good luck with your plants.