Think of it like email.

The email provider is the instance:

Google, Yahoo, outlook, etc

The community/user is your email address:

Android@gmail, Android@yahoo

Because they are different providers you can have the same name since it's the only one on that provider.

A subreddit can be uniquely identified like this /r/games. Once that name is taken, no other sub can have it.

A Lemmy community is identified like this: [email protected]. Notice it has a name part and a server part, both are necessary to uniquely identify the community. [email protected] is a different community with different mods and different posts/comments (unless someone crosspost, but then it's still two different posts with mostly the same text inside). You can mostly access any community on any server from your account, mostly irrespective of what server your account is on.

But similarly named communities on different servers are different communities. They are analogous to /r/DnD vs /r/dndnext. Similar topic, different subs, same with those Lemmy communities.

Yeah:

Instances = Servers

Communities /c/ = Subreddits /r/

Basically, yes. And it's not too much of a stretch to think of lemmy.world and lemmy.ml as two separate Reddits that share with each other, so anyone on either one can access stuff on both.

So two communities that both have the same name but are on different instances are actually two entirely separate places since they're on two entirely separate "Reddits". One is [email protected] and the other is [email protected].

Yes, essentially.

User U1 couldn't access anything, since their account is at hostone and hostone is down. But they could log in with a different account on a different instance (or just lurk without logging in) and be fine.

User U2 would be almost entirely unaffected, since User U2 is on hosttwo, which is unaffected. B@hosttwo would be entirely untouched, and User U2 would even still be able to access A@hostone. Sort of.

All along, when User U2 accessed A@hostone, they were never actually accessing A@hostone directly - they were actually accessing a mirror that's hosted on hosttwo - A@hostone@hosttwo. So the only effect of hostone bring down is that A@hostone@hosttwo wouldn't get any new content from A@hostone (or from any of the other federated mirrors - A@hostone@hostthree or A@hostone@hostfour and so on). But all of the content that was already at A@hostone@hosttwo would still be there and could (presumably) still be accessed. New content could (presumably) even be added there, but since it wouldn't be able to sync back up with hostone, it wouldn't show up anywhere else - it would just be at A@hostone@hosttwo.

And a note on those (presumably)s - internally, the lemmy/kbin/whatever software would recognize that it was failing in its attempts to sync with hostone, and likely that it was failing to even contact hostone. I don't know how the assorted pieces of software - kbin or lemmy or mastodon or whatever - handle that. If they ignored it and just kept trying to sync with hostone and failing, then User U2 might not ever even be aware of the fact that hostone is down, since even A@hostone@hosttwo would look the same - it just wouldn't be syncing with hostone, so wouldn't be getting any new content from there or from any of the other federated versions of A@hostone, and User U2 might eventually notice that.

It's also possible though that the software could be set up to tell User U2 that hostone was offline, and it might even be set up so that it would refuse to accept new content at A@hostone@hosttwo until it could get back to syncing content with hostone. I don't know why it would be done that way, but it could.