If you send an "offer peace" offer, the AI will demand what they want from you in the war. Even if you are winning the war, and send an offer peace, they will send you a peace offer they would like to give you: You will always get more if you manually send your demands, and you will always lose less if you manually send your suggestion. Suggesting an offer right at the start of the war means the offer sent will have to overcome the AI's modifier from war exhaustion, which is likely low or zero, any length of war modifiers, etc etc, so the early offers have to be really big to make the AI peace out early.
If you're the war leader, offering peace early essentially means you're surrendering without a fight, so the AI will try to grab all of what it wants. It declared the war because it's confident it can get it anyway, after all - why shouldn't it take the maximum of what it thinks you're willing to give? So it's neither of your two suggestions; it's simply the result of a nation having untouched armies, untouched lands, and a high willingness to fight on.