Does it show or ask for a stopper or show a real suit

I've had several auctions come up lately where there was some
confusion about where a bid was asking for or showing a stopper or
natural. It's easy when the opponents have intervened because bidding
their suit would ask for a stopper, in an auction such as 1S-2D-(2H);
3D-3H. And if the opponents have bid 2 suits, bidding one of their
suits tends to show a stopper in that suit but not their other suit,
such as 1D-(1H)-1S-(2C); 2D-(P)-3C.

Without an intervening bid, there is often doubt about when we bid a
new suit at the 3 level on the 3rd round of the auction when we have
not found a fit (or found a minor suit fit).

A) 1S-2C; 2D-3D; 3H

B) 1S-2D; 3D-3H

C) 1S-1NT; 2D-3D; 3H

D) 1D-2C; 2S-3H

In each of these cases, 3H is the bid in question. In A, it is clearly
a stopper ask or a cue toward a diamond slam. Responder initially
treats it as a stopper ask, with the expectation that opener will cue
bid again if he has a slam try. This auction came up on BBO last night
and we successfully avoided 3NT with Qx opposite Jx while 5 of either
minor rolled. (Sean bid 5C over 3H, which I think is weaker than 4C,
but it's also debatable about whether you can stop in 4 of a minor in
an auction like this.)

B is different from A in that there are two unbid suits, only one of
which you can bid below 3NT but I think it has pretty much the same
meaning.

C differs from A in that responder didn't game force so we can't rule
out having a 5-3 heart fit (with responder having 5). For most 2/1
bidders, there's no way responder could have shown an invitational
hand with 5 hearts so it kind of makes sense for 3H to be showing a 3
card heart suit.

Fourth suit when you're in a game force, as in auction D, is certainly
not artificial 4th suit forcing (like it would be in 1D-1S; 2C-2H) but
does it make sense to be natural? Opener has denied having 4 hearts so
unless you really like Moysians, responder would need to have 5 hearts
(and therefore 6 clubs) to bid hearts naturally. That seems unlikely,
and when opener has shown at least 4-5 in spades and diamonds, it's
likely a total misfit deal. I think a more productive use of 3H here
is to ask for a heart stopper - something like Axx, xxx, Qx, AQJxx.
Then again, that hand can temporize with a slightly misleading 3C or
3D bid. This one came up Tuesday night and sure enough, Emory has 5
hearts and 6 clubs and we avoided a good 3NT because I didn't bid it
with stiff Q. 5 of a minor also rolled but lost matchpoints to all of
the 630s in 3NT.

These are interesting things to consider and I don't think I have any
definitive answers but a rule of thumb is that if it is possible to
still have an 8 card fit in that suit (without having a real freak
hand), it should be natural. If we could confidently rule out that
suit as a place to play, it probably ought to be stopper ask.