I'm not sure what he told me exactly. Certainly he started to say things like I was "mean"; he might well also have said I'd been unfair all along. Altogether, it was a rather unfriendly rant that made very little sense.
In any case, though, I actually returned to a hard save I had made quite some time before, just to see if I could change things. I started the quest again, did exactly what I had done before (again, 15 moths - arghh!), but made sure to spend some more time on the sphere/bow game. That helped - in the end, Rynéus trusted me. If nothing else, it made me feel a little better, even if, afterwards, I had to return to the "bad ending" version to avoid having to play through all the black stone quests (and a few others) again from beginning to end.
So I suppose Rynéus's first game did make the difference. I'm still not entirely sure why, though. It could be that I played it a little longer than the first time round. It could also be that this time, I made sure to hit the sphere (I believe) three times in a row. In any case, I clearly failed that game the first time without ever realizing it.
Thanks for your reply, by the way!