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More precisely, what needs to happen to win a war? In other words, what triggers the AI to offer peace, with or without gifts?

I'll begin with what I've observed.

First, it appears that intimidating a foe - in other words, threatening to cause lots of damage - is the way to get them to offer peace. Wipe out their attacking force. Retake or destroy an equal amount of what they took. Then send a large invasion force toward their colonies after destroying its defenses.

But don't actually take it. Threatening to do so makes the AI want to stop the fighting.

If you destroy a little too much, it seemingly makes the AI want to get even. This is especially true when the player initiates war and the AI is stronger by the numbers. If it gets to that point that too much was destroyed, they'll only offer to give up when they've been effectively crippled- they won't admit defeat until they've been thoroughly defeated.

Then it appears that the unspoken cause of war matters a lot. An unjustified war by the player pushes the AI to fight until defeat. An aggressive AI that declared an unjustified war will give up if they fail to take what they wanted. Those that join a war due to a Defense Pact or a triggered event are willing to end the war without a single shot fired in time. Then provoking an AI to declare war on you is even harder to discern to what point the AI will be willing to give up. "Please don't completely crush us, we'll pay you to stop."

Another way of looking at the war is to consider what it takes to offer peace- a war could be seen as forced diplomacy. "I wanted that colony. You wouldn't trade for it. So I took it. I'll offer to buy it for a fair price. This large amount is enough, let's stop fighting now."

The War Score is obviously nonsense in this regard. It can be astronomically high and the enemy may not be willing to give up. The War Weariness contributes to crippling an empire, but like the War Score, apparently doesn't dictate how likely they are to give up.

Any insights (or arguments against what I've observed) would be appreciated!

I have probably watched this and come to roughly the same conclusions. I think it is quite refreshing that the AI sort of thinks like the player normally would. If the AI declare war and manages to early on take one of my poorly defended colonies I'm not really in the mood to agree to peace until I at least exhausted allot of resources to retake it. The AI seem to do the same thing.

All tough loosing some major battles and then see a threatening invasion force closing in on one of your planets would gladly persuade me to sue for peace even if I lost a mine or two... or even for giving them a mine or two in compensation for the peace.

Different races also seem to be more or less willing to negotiate peace... a Mortalen empire that feel strong is very difficult to get a peace deal from... you really need to pound them hard first. While the Teekans seem to be more willing to make peace after some time, even if they have not really lost anything and still have a strong fleet. War simply is not good for business I suppose...

In my last game a weaker Mortalen empire was in a perpetual war with a much stronger Boskaran empire until most of their colonies fell, they simply refused to give up even when being fully conquered. Once I intervened in that war by sending war fleets to protect the remaining Mortalen planets could they finally muster strength to start reclaiming their lost colonies... this war probably lasted more than fifty years and ruined both empires economies in the end.
I'm pretty sure the Mortalen in this case would have rather fallen on their sword than make peace with the Boskarans.