I don't agree with all modern expert philosophy, but doubling with good 13 or 14 3-3-4-3 is one I do agree with. Passing might have worked ok in the days where responder routinely passed with <6 HCP, but those days are long gone. Now you can easily get ‘frozen out’ with combined 26 HCP (or more).
Also, at Nobody vul, there are tons of ‘small imp gains’ from getting in. You go -50 in something instead of -110 or -120. Or you go +110 instead of plus 50.

We'll have to agree to disagree.

Mike M: “Can advancer have an invitation hand without both M with the cue bid?”

Yes. He can also have a hand where, if our fit was in a M, he would play game, but since it's in a minor, he's willing to go a little low.

Ray: If, in Michael Shuster's example elsthread, the player who bid 6♠ realized his error before opponent acted, I see no harm in letting that be changed.
But I think your suggestion is fine also - certainly compared to the current Law.

If so, then now bid 4♥ over 6♥?

In Scotland in the 1970's, a common joke was "What do you call with jack-to-eleven and a yarborough'.

Passing with the OP hand is wrong. So is 6N. But no more than 1♦, 2N, 4♣ and 5♥ were wrong.

Without too much analysis, I think the rule should be ‘before the next player calls’.

Even more important, you might consider giving a bridge hand.

Steve: It looks to me that the winning defense(s) are easier against the ♥ finesse line. Let's say I play ♣K and LHO has Ax. Even though the 52% in clubs didn't come home, West might fear that the hand is such that continued trump plays will allow me an eventual ‘lock’ by later running ♣10. So he might switch.

Furthermore, against weaker opponents, if RHO has the doubleton spade I might actually run ♣10 pitching a HEART - maybe LHO will not find the ♦ shift from the K.

Also, there's still the chance of ♥Qx.

Another though is that after winning trump, ♣K (they win), second trump, ♣Q I play ♥AKJ ruffing in dummy and ruff ♣ If LHO has (say) xxx, xxxx, Jxxx, Ax I still make. I cash heart and play a ♦ and they can't stop me from separating trumps. I guess there may be variations on this theme.

When I first saw this hand, my thought was ♥ finesse might be percentage, but I think I'd play on clubs in practice. I still haven't attemprted any detailed analysis (and won't).

Ed: I have never Directed. I admire Lord Denning.

“Not sure why if responder takes a delayed 3♣, why that should be a problem. Advancer will X.”

Does advancer always double, or only with a GF? Any GF? Since passing is forcing, is double penalty? Does any partnership know this stuff?

What happened was that advancer, who had an invitational hand (I think he was 4-4-2-3 with a ♣ stopper), passed. Doubler with a minimum and 4-card ♥ bid 3♥. And now advancer was understandably unwilling to pass so 4♥ doubled down 2 (or 3?) was reached.

Delayed actions often cause problems, because we have little experience of them. I have a page in my notes with some agreements, but it is a bit ‘sparse’.
But seeing the above result made me think, I'm going to bid 2M over the cue if I have a M'.

“You must have a 4 card lengh in ♦. (with 5+ you had bid 1♦, right?)”

Not definitively. If I'm (say) 3-3-5-2 with poor ♦ I might prefer double.

“You dont have a 4 card M. And you are not willing to bid 2SA or 3♣. Such a Hand does not exist for me.”

If I have KQx, KJx, A10xx, xxx I'd definitely want to double. And I could have a little more. What do you call over 1♣ with hands like that?
If I need to bid 3♣ on such hands what does advancer do with diamonds and a ♣ stopper (Axx)? If he bids 3♦, what do I do now?
So much easier to start with 2♦. Then we have room to let asdvancer describe - and can still get out in 2N or 3♦.

Enjoy your 3-3 ♥ fit. you might have had a 4-3 spade fit.

Mike M: The drawback of 2♦ with a M is not as great as the drawback of (say) bidding a 5-card m instead of 4-card ♠ after (1♥)X-(P)2♥, (P). But it still exists. If advancer bids (say) 2N over 2♦, then doubler, with a minimum, is ‘caught’ - having to guess which major advancer has.
I guess if you agree that advance ‘must’ bid up the line over 2♦ you could solve that problem. But if the 2♥ bid could be 5(+)-card GF OR ♥Jxx with a 1c stopper, it's going to be tough to sort that out. The bids carry less meaning in terms of suit quality and interest in NT or slam.

One thing I once saw was when responder, surprisingly, took a delayed action 3♣ bid over 2♦. The partnership could no longer play 3♥ but was forced to get to game.

Everything I've seen has made me think it's ‘simpler’ to bid your M first. Nothing bad as happened (at least yet). The opposite is not true.

Of course, if I have longer ♦, 4-card M AND slam seems in the picture, I'd start with 2♦ and bid the M next - and MY partner wil know what I have. Tours won't. But if I doubled on (say) a (4-3)-5-1 minimum, I bid my M.

As regards, ♦ slam hands, one (non-Standard) agreement I like to have is that raising 3m to 4m is forcing when both are unlimited.
It's not that playing raising 2♦ to 3♦ as GF and bidding up the line is unplayable. But you would need an agreement for this, and I've never heard of an expert partnership having one.

Buddy: “2♥ not 3♦ by North is much better.” Perhaps, North thought a cue bid followed by a new suit was forcing, maybe even game-forcing? I must admit that I might think so”

David Wetzel has it right. The ‘rules’ (which used to be Standard) in my notes are that advancer's cue bid is forcing until a suit is bid and raised OR 2N by advancer.
It doesn't matter who bids the suit first.

Buddy: I actually thought of that hand type and decided I would bid 3♣ which is GF and perhaps even suggests this hand type or 4-4-4-1. If partner (sadly) fails to bid 3N I will need, without special agreements, to follow with 3N. Partner would have to remove with a stiff club.

“I suspect that bidding after a take-out double and cue bid has not been completely discussed by many partnerships.”

You are correct. And there is much that should be discussed.

“I'm starting to have sympathy for both players.”

You shouldn't. Making a NF raise of a possible/likely 4-card suit to the 3-level is ‘non-bridge’.

Ray: First, that Law should be changed. Second, after the BIT I could not be convinced that the original call in the OP was unintended.

Wendelin: “Why not showing your shape and stop with 2SA…”

Perhaps, because, as a bridge player, you'd rather make an effort to have partner be declarer, with holdingsa such as KJ, or Qx, or Qxx.
Even on the actual hand, you're better off declaring from partner'side. It's usually desirable to put the strong hand on lead.

On the contrary, I think it is NORTH who should have bid 2N over 2♦. The lack of a club raise by responder combined with the lack of a double of 2♣ by opener makes the half stopper more likely to be viable.

“the 2♦ bid should be game forcing anyway.”

I see no logical reason why. If you had bid 2M, advancer's raise to 3M would be a NF invite. It doesn't maker sense for us to be in a force when our only 8-card fit requires us to make yet another trick to score game there. It's pretty easy to construct combined 26-28 HCP hands that have no reasonable game available.

Raising a possible (for me, likely) 4-card suit to make a NF bid with a doubleton and maybe play in a 4-2 fit is not (as Patrick would say) my cup of tea. 3♦ would be my SIXTH choice.

Richard: I used to do that too. But my experience, both for me and others, is that it's simpler to uncover the M fit as fast as possible.

“They might, but I hardly think they'd keep the recordings for over 3 years.”

You never know unless you try. What's the harm in asking?