I'm skeptical that either of the solutions you've settled on are the right ones - both, for me feel a bit broadcast mode. When I read your post I worry about the level of trust and psychological safety between you both, so thats what I'd be working on. I think you'd be better served having open two-way chats about work; sharing fears, hopes and dreams as a start.
I get its not always neat determining how things start - as others have said there are a few smells there. To be blunt - its unclear to me if the other person in the relationship will 100% see it the same way as you do in the partner vs co-founder thing. You think there is a problem with terminology, but think he's got the same definition of co-founder as you? I'm skeptical - but only you can truly appraise what type of relationship you have and if the distinction really matters to you.