a very broad definition would include all of these social networks - most of which are incompatible - but in the most common sense these days, people mean anything ActivityPub compatible, with Mastodon being the most popular server software for that. i suspect Nostr may have taken some inspirations and lessons from it, but wanted to make it more decentralised. on Mastodon, you're still relying on making accounts on specific servers. the ability to migrate identities was added a while back, but you can't bring your posts with you, so it's still a bit limited.
Discussion
these = https://fediverse.party/
Does mastodon have clients like Nostr?
plenty of them. it's been around since 2016 and has about 1.8 million active users.
member.cash or mastodon one o them?
can you cross an account to nostr account to fediverse?
not familiar with member.cash. the top servers are these. https://void.cat/d/9PukEcFTVMVVaqsKaKecF1.webp
it's hard to recommend a specific server, but what people usually do is sign up for something like mastodon.social, mastodon.online or mstdn.social and then later they find a permanent home elsewhere. i have a server at berserker.town myself.