There is a lot of debate about this in the community. People like pvh, adam wiggins, and johannes have all said they consider Replicache and Zero part of the local-first movement [1] and that they would be sad if we and other things like us weren't in it. I was even invited to speak at local-first conf, where we announced Zero last May. [2]
I think in practice delivering on the local-first ideals is ultimately up to the app developer not the library. Like in practice every yjs app talks to its server over some bespoke protocol and is loaded over a URL. If that server goes away your yjs app is a brick. No less a brick than if your zero server goes away.
Similarly although everyone likes to claim Linear as a local-first app, if the Linear server goes away your Linear app is a brick. Yeah you can export the data from IDB or whatever but you could do the same for Zero or any of these systems.
That all said, for our part, we have recently tried to avoid claiming Zero and Replicache as local-first on our own sites to avoid this controversy [3].
I think we're building something really valuable that a lot of people are excited about. And I think we're making the right tradeoffs for the majority of software that most people use every day. So I don't really care too much if it qualifies as local-first (except insofar as I like hanging out with the local-first folks and would be sad to not get invited to the parties).
I forgot to mention this to Nate when he showed me these mocks for One because honestly almost everyone else (not us) refers to Replicache and Zero as local-first and it's easy to forget. I'll leave it to him to decide what to do with his own product, but it seems easy enough to me to soften the language.
[1] https://youtu.be/cgTIsTWoNkM?si=6bYAAunqd5kdl-UL&t=2692