Your first looks like a Rat Tail cactus, you can read about it here https://www.thespruce.com/aporocactus-flagelliformis-definition-1902538

The second could be a mammillaria elongata cultivar "Copper King", tell me if that looks right https://llifle.com/Encyclopedia/CACTI/Family/Cactaceae/11618/Mammillaria_elongata_cv._Kopper_King

It can be quite hard to ID plants if the light is behind them but yours look good, the new watering schedule sounds like you are doing it right. The right word for the fronds would be stems 😁 Cacti mostly don't have leaves