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We figured out Where and when did "the GM is always right" get codified first? to the early days, or more precisely, it predates the main bulk of AD&D and can be pinned to have shown up in the 1970s.

This question is about the general rule/goal Have Fun. We find it nowadays in a lot of games, expressed in various ways. Some examples:

Exalted 3 (2016), p24:

How to Play This Game [...] Unlike most games, there’s no fixed way to “win” Exalted.
The goal isn’t to advance your character to some ultimate
victory-point; rather, the goal is to have fun telling an engaging
story with your friends.

Gurps 4th Edition - How to be a GURPS GM (2014), p4 (emphasis as in print):

Introduction [...] There is absolutely no One True Way, no “official” way,
of running or playing GURPS! The whole purpose of the
rules is for everyone, the GM and players alike, to have
fun, no matter how they do it.

Dragons at Dawn (2010), p30:

Arguing [...] Now if the players are arguing with the Referee, everyone should keep in mind that the point of playing games is to have fun. Rules be damned; if no fun is being had then work together to change things so everyone is having fun or else go watch a movie.

It does not suffice to have "Have Fun" called out as a goal when trying to explain a different point, it has to be able to stand alone like in the three above examples. A non-suffiient example would be Basic Dungeons & Dragons Dungeon Masters Rulebook(1983), p2, that does not demand that fun was the overarching goal of the game but merely uses it as an illustration how fairness was to be expected:

There is one rule which applies to everything you will do as a Dungeon Master. It is the most important of all the rules! It is simply this: BE FAIR. A Dungeon Master must not take sides. You will play the roles of the creatures encountered, but do so fairly, without favoring the monsters or the characters. Play the monsters as they would actually behave, at least as you imagine them. The players are not fighting the DM! The characters may be fighting the monsters, but everyone is playing the game to have fun. The players have fun exploring and earning more powerful characters, and the DM has fun playing the monsters and entertaining players. For example, it’s not fair to change the rules unless everyone agrees to the change. When you add optional rules, apply them evenly to everyone, players and monsters. Do not make exceptions; stick to the rules, and be fair.

When was the first time a clear "Have Fun" was put into a game's rule book as a leading idea that one should achieve for all the group?

The earliest possible encouragement to have fun appears in the foreword to the Original D&D Men & Magic (1974):

With this last piece of advice of advice we invite you to read on and enjoy a "world" where the fantastic is fact and magic really works!

As mxyzplk answered, Swords & Spells (1976) includes a similar exhortation in its foreword, referring specifically to the DM's right to change rules to improve the game:

Finally, as with any set of rules done by any author and any publisher, these are for your playing enjoyment. If you find sections which do not bring you enjoyment, alter, expand or delete them as you see fit. Be careful, though, so as not to destroy the flavor of D&D!

The adventure module In Search of the Unknown, included in the 1977 basic set, makes the explicit reference that the Dungeon Master is responsible for the players' enjoyment:

The Dungeon Master, as referee, is the pivotal figure in any game of Dungeons & Dragons. Accordingly, his ability and expertise — as well as fairness — will be important factors in whether or not the game will be enjoyable for all the participants, as well as for himself. ... His responsibilities are considerable, but his foremost concern should be to provide and enjoyable game which is challenging to the players.

It also includes this specific advice in the tips for players:

Enjoy yourself, and good luck!

The 1977 Holmes Basic rules themselves make one of the first references to tbe exact word "fun", again in the context of the DM changing rules:

The game is intended to be fun and the rules modified if the players desire.

Later, in the AD&D 1st edition Dungeon Master's Guide (1e), Gygax formally advises in the section "Approaches to Playing Advanced Dungeons & Dragons" that the goal of the game is to have fun, and he uses the word fun:

Advanced Dungeons & Dragons is first and foremost a game for the fun and enjoyment of those who seek to use imagination and creativity.

In this context, he's defending D&D's perceived lack of realism, its weaknesses as a simulation, by defending, as Gygax wisely often did, that D&D was a game intended for fun rather than to simulate the most precise outcome. By 1979, other RPGs had been created which attempted to be better than D&D by being more detailed or more realistic, and the result was often an incredibly tedious RPG that was no fun to play.

It Has Always Been There

RPGs are games. Games are for fun. The earliest RPGs say so - in the OD&D Author's Introduction in Swords & Spells (1976), Gygax closes by saying, "Finally, as with any other set of rules done by any author and any publisher, these are for your playing enjoyment. If you find sections which do not bring you enjoyment, alter, expand, or delete them as you see fit."

(I trust we don't have to debate whether "for your playing enjoyment", when translated from Grand High Gygaxian, means "have fun.")

The Confusing Admixture Of Have Fun And Rule Zero

The real genesis of your question here is from the deleted question about "Rule Zero - is it have fun or the GM is always right?" What this and similar quotes indicate is that that's a false dichotomy.

The GOAL of a game is to have fun, for "your playing enjoyment." That's not a "rule," that would be weird.

As RPGs are games with rules, and the books you get to start playing an RPG are full of rules, while those rules bring you fun they are also the first potential threat to your fun. Therefore even from the very beginning, those games have noted that you should feel free to modify the rules to serve the goal. This was mentioned as "The Most Important Rule" in Basic D&D and as Rule 0 in 3e D&D. This popularized the term, so you can for example see the TVTropes page for "Rule Zero" for citations. It's truly the underlying rule (all these rules to come after me, don't take them too seriously).

So why the confusion between Rule Zero and the goal of Have Fun? Well, like most disinformation on the Internet, probably 4chan. The Rule Zero page in 1d4chan (not linking to it per site rules on linking to hives of scum and villainy) says

Rule Zero is supposed to be the most important rule of any RPG or miniature wargame. It's called the "zero-th rule" because it is so fundamental that it comes before even the first rule stated in any rulebook. It's also a meta-rule, so it doesn't really fit into any numbered lists of rules. Good RPGs and miniature wargames will mention it or something like it in the rule book(s), usually something wishy-washy like "these rules are only a guideline." Sometimes this rule is confused with the Golden Rule: "whatever the DM says, goes." Suffice to say, That Guy is not a fan of either of the above.

Definition

Rule Zero:
"Roleplaying games and miniature wargames are entertainment; your goal as a group is to make your games as entertaining as possible. If that means breaking the rules temporarily, or permanently as a house-rule, then so be it."

Note that it instead relegates GM decisions to some separate "Golden Rule" thing whose page, of course, cites exactly zero sources of a game calling that the Golden Rule, just as the citations on their Rule Zero page fail to support their definition well here. To be fair, the definition has two parts - "the goal is to be entertaining" and the corollary "that means changing the rules is possible", but italicizing the first and separating it from their "Golden Rule" causes confusion, IMO, and is probably where someone saying "Rule Zero, isn't that Have Fun?" comes from. It doesn't come from any game book. While many games say their point is to have fun or their goal is to have fun, they don't state it as a rule and certainly don't state it as Rule Zero because that's, you know, a thing that already has a definition.