Which South hand fits the bidding best?

Solution

South 1 has no reason at all to bid 2♦. He can describe his hand perfectly in one bid and therefore should do so: an invitational 3♥ bid*! In not doing so, he denies this kind of hand!
Now that South 1 has not bid 3♥ on the first round, he can't bid it on the second, since now it would be forcing (for an explanation, see South 2).
* Often called a 'limit bid'. This phrase describes the bid less clearly, since there are many limited bids, but not all of them are invitational.

South 2 wants to be in game. But he doesn't know which one. It should be 3NT if North has only four hearts and 4♥ if North has five.
So South 2 is the one to bid according to the given auction, giving North a choice between 3NT and 4♥. Therefore it's logical that 3♥ is forcing! With a hand of invitational strength South would have bid 3♥ directly (with a four-card heart suit, like South 1) or have passed 2NT (with less than four hearts; if South has three and North five, NS thus miss the 5-3 heart fit — and possibly 4♥, since in view of the fit North and South each could have added distributional points; this is a disadvantage of opening four-card majors).

South 3 has the perfect hand to bid 4♥ on the second round (2♦ was correct). After North's 2NT bid, he is known to have: (12)13-14 HCP, a four- or five-card heart suit and a balanced hand. So South 3 knows that there is a heart fit, that NS have sufficient strength for game and certainly too little strength for a slam. So South 3 simply bids the right contract.
Why didn't he bid 4♥ on the first round of bidding then? A long time ago he would have, since such a direct raise to game in major suit used to show 12-15 HCP (in beginner's courses this is still common). However, nowadays such a direct raise (1♥-4♥ or 1♠-4♠) is pre-emptive. Therefore South 3 first bids 2♦ and next, if partner's rebid shows little or no extra strength, he jumps to 4♥: a delayed game raise.

PS: In the text above we assume that NS do not play the Jacoby 2NT reply, showing four-card heart support and values for game or better. South 3 qualifies for such a bid.