I’ve heard of the term “suing for peace.” The Wikipedia article is a stub and doesn’t explain too much. Is there an actual court involved?
I think you’re letting the legal practice of suing someone (taking them to court) cross over into other meanings of the word. Consider: Suing Definition & Meaning - Merriam-Webster in particular the first example given for it’s use as a intransitive verb and it’s attached example.
“Suing for peace” is basically asking your enemy “if you want me to surrender, let’s talk about the terms.” You and your enemy can then haggle over everything from establishing your boundaries to prisoner exchange to whether you get to keep your government intact.
Of course, those discussions can break down, or your enemy can always say, “there are no terms. Surrender or be destroyed.”
Even in the legal sense it has this meaning. When you sue somebody, you’re actually petitioning the court. Of course, in common parlance we say “I’ll sue you for every cent you own!” instead of the more accurate “I’ll sue the court for every cent you own!”
While ‘suing someone’ is a case of pleading the court not all pleadings and requests are legal proceedings in court. It’s akin to how apples are fruit but not all fruit are apples. The OP’s question about their being an actual court involved is probably confusing the two. Taking the more specific (pleading to the court) and applying it to the general (pleading). That is what I meant about confusing the two.