Oddbean new post about login | logout I think inbox/outbox are the most well understood terms. Other people put things into your inbox, and so you need to check it for new things. You put things into your outbox and everybody comes to see what you have posted. Of course you also put things into other people's inboxes and you read things from other people's outboxes, so clients have four kinds of paths (actually 5 since reading from other people's inboxes lets you see full thread replies). Gossip also has an unpublished outbox called "write" which instructs gossip to always write to this relay (in addition to the norm) whenever it writes. Gossip also has an unpublished inbox called "read" which instructs gossip to always read from this relay (in addition to the norm) whenever it is seeking something. These terms and usages might be confusing and I'd take feedback on alternate names for these.
I think "inbox" is a no brainer. Everyone understands what it means. Outbox is not as easy because email clients never made "outbox" a real thing. For that, I do think "Home" could be a better name because it suggests that this is the relay that has your posts for others to see, your home on the web. And you should be very careful on the relays you select to be your home. "Getting posts from people's outbox" is a weird thing to say. And "Getting posts from people's home relays" is better understood. FYI, 1. I use kind 10013 to your "unpublished write" relays, which I call Private Home as described in the Generic Draft NIP: github.com/nostr-protocol/nips/pull/1124 2. Your private inbox to me is not an inbox. It's either an index relay (like purplepag.es) or a new kind that could lists large relays on the hopes of finding events the client can't find anywhere else.