I realize 'read' and 'write' are confusing because it depends which side is doing the operation. But if we change it (by adding "outbox" to mean the same as "write") then many clients who haven't coded for the change yet will get relay lists they don't understand. I think it is too late to fix this aspect. I wish we could.
I agree we need both public and private relay lists. NIP-65 is the public one. It is not intended for configuring your own client, which very likely needs more information, even client-specific information.
I don't think a home relay is sufficient because there are plenty of more complex situations it wouldn't cover. For example, I may post to a paid relay, but I cannot expect people to copy me in on a thread at a paid relay that they may not be paying for.
IMHO NIP-65 is finished and stable. The only thing that makes sense to me to add is a DM marker for where people should send you DMs (some people might want that to be a more secure relay that only swallows events and doesn't serve them).