Tough problem. Agree with Mike that partner's got to have a diamond control. With the fifth trump and only five minor suit cards, I think I'm just barely good enough to bid slam.
I would also respond 1♦ with this hand, playing Walsh. I'd always respond 1♦ with GF values and a good 4-card suit. We had* methods to distinguish between balanced and unbalanced hands with a 4-card major after responding 1♦. (*I've played T-Walsh for years now, where we responds 1♥ with this hand.)
The first one is, at best, a marginal game. I'd not be disappointed for missing it.... nor bidding it. The auction was fine by me. The second one is a terrible game, even the 3-level might be too high. Better stay out of this one. Fine auction.
I don't think the hesitation points in a clear direction as to what partner was thinking about. To me this is a clear raise to 5♦. With the hesitation I'd go out of my way to avoid making a bid that might look like catering for some possible hands that might have caused partner trouble. Thus, 4♥ is out of the question. I bid 5♦ also after the hestitation.