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
Thank you!
these are both the same link though
Oops, fixing that now
They are both definitely in the same genus of the ones I have, if I remember the science I learned 60 years ago correctly. M copper colored one is copper colored all over, it's very beautiful and I am delighted that it could bloom! Ang my rat tail looks like the first picture in that link, not so much like the second, mine is not so fleshy..
Nice! Copper king is very easy to bloom and the Rat Tail has spectacular flowers
I am so happy! A friend of mine was getting rid of a huge succulent collection, and I was the first to choose. I don't remember every falling for a cactus before, but I fell for both of these.