I can't address the entirety of the Mind Game, so I will just deal with your more specific question: the symbolism/meaning of the Giant's Drink. For Ender's Game I have the paperback Author's Definitive Edition (1991), and for Speaker for the Dead I have a paperback (not sure which edition). Both are from Tor publishing.
Ender's Game online table of contents, and Speaker for the Dead online table of contents (scroll down and expand)
First, let's establish what the Giant's Drink is intended to be by the adults who run the school. It is supposed to turn children away and make them give up. Surrendering in the face of overwhelming odds is seen as the good, rational decision.
But the game had no victory conditions--no matter what the child did, his analogue died a gruesome death. The human psychologists measured a child's persistence at this game of despair to determine his level of suicidal need. Being rational, most children abandoned the Giant’s Drink after no more than a dozen visits with the great cheater.
Speaker for the Dead, Chapter 11: Jane, page 192
Ender himself knows that the Giant’s Drink is an impossible game. Despite this, he feels an obligation to try it, and is unable to keep away and call it a loss. He doesn’t make the expected, “rational” choice of giving up.
Not through the mousehole this time, he told himself. I'm sick of the Giant. It's a dumb game and I can't ever win. Whatever I choose is wrong. But he went through the mousehole anyways.
Ender’s Game, Chapter 6: The Giant's Drink, page 62
The game itself is a representation of how the adults who run the school can’t fathom any options besides the binary choose how to die/choose to give up. Additionally, they expect every child who attempts the game to accept that there are no other options, without question.
the Giant only played the guessing game. Stupid computer. Millions of possible scenarios in its memory, and the Giant could only play one stupid game.
Ender’s Game, Chapter 6: The Giant's Drink, page 63
But Ender does not accept these assumptions. The adults who created it only saw two choices, Ender attacking the Giant, then, is a clear repudiation of the rules and boundaries the officers try to set.
And instead of pushing his face into one of the liquids [the choice he's supposed to make], he kicked one over, then the other, and dodged the Giant's huge hands as the Giant shouted, "Cheater, cheater!"... and as the Giant screamed, Ender's figure burrowed into the eye
Ender’s Game, Chapter 6: The Giant's Drink, page 64-65
He tried to get his onscreen analogue to do outrageous things, things not "allowed" by the rules of that portion of the Fantasy Game... finally, one day, the boy surpassed the program's ability to defeat him
Speaker for the Dead, Chapter 11: Jane, page 192
But there’s more to Ender’s choice than it simply being unexpected by the creators of the game. Ender considers his third option immoral, even though it was the option he chose. He is disgusted with himself for resorting to gruesome violence. He feels betrayed, as it “was supposed to be a game” - the Mind Game, living up to its name, suddenly becomes very real to him through how his actions affect his mental state. To win, he became ruthless. Just like his hated older brother, Peter.
He had made it. He ought to explore. He ought to climb down from the Giant’s face and see what he had finally achieved.
Instead he signed off, put his desk in his locker, stripped off his clothes and pulled his blanket over him. He hadn’t meant to kill the Giant. This was supposed to be a game. Not a choice between his own grisly death and an even worse murder. I’m a murderer, even when I play. Peter would be proud of me.
Ender’s Game, Chapter 6: The Giant’s Drink, page 65
To recap: the Giant’s Drink
- Is expected to be unwinnable
- Prompts Ender to make an choice not expected by the creators of the game
- Leads to Ender being disgusted with himself for making that choice
Therefore I would say that the Giant’s Drink, and especially Ender’s attacking the Giant, symbolizes his unorthodox decision-making. He at the same time embraces a cold, ruthless side of himself to win the game, and is disgusted with himself for his action within the game.
Hmm… that sure sounds like another part of the book, doesn’t it?
Or multiple parts. The lessons of the Giant’s drink serve as the main symbolic parallel for the whole book. I’ll list the parallels in chronological order through the book.
In Chapter 1 Stilson, a bully, challenges Ender to a fight. No adults are around to protect him. There seems to be no good way out. Ender makes a completely unexpected choice and not only defeats Stilson but destroys him, beating on him even after Stilson is down for the count. Then he leaves feeling disgusted with himself for such violence. He knows that what he did went against the etiquette for such fights, but he felt that he had to do so to win.
In Chapter 4 Bernard harasses Ender while the two are in the shuttle. Again, no adults help. Ender again goes for the “overreaction” and pulls Bernard out of his seat, throwing the boy across the shuttle. He is again surprised at himself for how much damage he does, though in the moment he saw it as the only way to “win”, to stop the bullying then and there.
In Chapter 12 Bonzo tries to hurt, and possibly kill, Ender. Ender again does the unexpected and beats Bonzo hard enough to kill. Though he doesn’t know that Bonzo dies at the time, he knows that he used more violence than necessary and is, again, disgusted with himself. He is aware that the “rules” for such a fight do not allow for such damage as he inflicted.
Right after that, still in Chapter 12, Dragon Army is forced to fight against two other armies at once. The traditional means of winning (take out all enemy soldiers, and then pass through the gate) is no longer viable. So Ender breaks the rules again. With the idea that “the enemy’s gate is down” he sacrifices the majority of his army to distract from a few sneaking to the gate. The fact that the adults broke their rules by pitting two armies against one, and the trauma of his fight with Bonzo, leads to him giving up on the game, on the school, on the future where he is the star commander of the Bugger war.
And finally, in Chapter 14 there is the climatic battle. The final battle, at the Buggers’ home planet, is expected to be impossible to win. The adults running the program can see no options that result in victory. But Ender makes an unexpected choice - sacrifice every ship just to get close enough to the planet’s surface to deploy the Little Doctor. With that he oversees the (almost) total genocide of a species. When he finds out what he has done, he is horrified with himself. He didn’t want to win if it came at such a high cost.
In none of these situations does Ender do the "rational" thing and give up. In all of them he finds an unexpected path that he hates himself for taking. The Giant's Drink is the epitome of the idea that while taking a third path may be necessary to "win", it will still hurt to go against one's morals.
The Giant’s Drink is the main symbolic parallel for the entire book. Over and over again, the motif of Ender being forced into a rule-breaking choice and then regretting it deeply comes up. Over and over again Ender finds a loophole and is disgusted with himself for doing so.