Perfect Misunderstanding

Most experts are heavily discussing their system with their partner in order to avoid any sort of misunderstandings.

On this important session in the league, a player was ill, and a sub was called to fill in. They had a long talk agreeing the conventions, carding and whatever else they could, including puppet, Bergen , DONT, 0314 and more. When they reached the issue of 2♣ openings they agreed that it should be game forcing in general except for the 2♣ - 2♦ - 2NT sequence where:
"2NT is 23-24, and you may pass with 0-1 points," said one.
"So, with balanced 25-27 we need to jump to 3NT," said the other.

Thinking they covered all their options here, they moved on to discuss other issues.

During the play, this hand appeared, on which they were the only ones to bid and make the following slam:

Dealer South, all vulnerable.


Dear readers, do you understand the bidding above? No? Don't feel too bad about it as it seems that the players themselves didn't get it either... Actually they both got it so wrong that they found the perfect way to a perfect contract, due to a perfect misunderstanding. Playing with screens, they were asked to explain their own bids and their partner's bids, and here is what they came up with:

Opener:
2♣ - Strong, artificial.
2♦ - 0-7 points, artificial.
4NT - 28-30 points ("Yes, we just agreed that 2NT is 23-24 and 3NT is 25-27, so 4NT is definitely 28-30.")
5♣ - Puppet, stayman.
5NT - No 5 or 4 cards major.
6♣ - Partner prefers slam in clubs.
Pass - OK by me

Responder:
2♣ - Strong, artificial.
2♦ - 0-7 points, artificial.
4NT - asking Aces, 0314.
5♣ - 0
5NT - asking Kings.
6 ♣ - 0
Pass? - "I wonder ... He cannot pass here."

As you can see, 6♣ is unbeatable: Having 11 top tricks, declarer just needs 1 diamond ruff in dummy. On a spade lead, he played ♦AK, ruffed a diamond with the ♣2, cashed ♣QJ, entered hand to cash the ♣AK and claimed 12 tricks, giving up 1 diamond in the end. All other pairs were in 6NT, going 1 down.

So... how do you think the bidding would have gone with his regular partner? ...