I'm sure I've muddied the waters, so I'll go back to work. :)

No, you haven't - and that's how I always thought of "Gen", too. But fandom definitions do change, and evolve, so I'm not so sure that's what it means to others any more.

Ship just means relationship. In some fandoms it only means het relationships (and then it covers all ratings, it just means the fic features the pair in a romantic or sexual context); in others it means anything romantic or sexual, het or slash.