If ID is correct, you probably need more sun to stop the branches growing so thin and that may be the reason for the lime-green colour (not normally expected on a cactus). Search Google images for other examples of how your plant should look (bear in mind that others may have their plants in lower light so some may also look thin and brightly coloured).

Bergzivman, at first glance I thought a Gymnocalycium, but the more I look the more I see a Lobivia;). Sorry not much help, but that's how cactus can be. Sometimes you just have to wait to see what the flowers look like. In this case you would only have to wait for buds to make a ruling if it's a Gymno or not. Gymnocalycium buds have zero spines, fuzz or anything. They're just smooth with a scale pattern;). Very distinct.

1)Opuntia something. 2) Cylindropuntia something Where do you live and are you growing these outdoors all year long? try posting on the Cactus & Succulent forum, not the C&S Gallery , but the actual forum.

Top is an Opuntia. Looks like an Opuntia I see at Home Depot that’s called Johnson’s hybrid. Pretty sure I’ve got the same cactus and that’s what the tag said Johnson’s hybrid. Not that that helps any. A hybrid between what Opuntias? Bottom is some type of Cylindropuntia. Looks to be Austrocylindropuntia cylindrica. The flower looks right for that id.