mdd — Actually Dovid Hamelech gave over Shaul’s relatives to rectify a Chillul Hashem that had occurred previously and was causing a famine in the land. (The Urim V’Tumim told Dovid the reason.) The Chillul Hashem was giving the impression that Jews do not keep their deals (because 7 Givonim were killed in the city of Nov, and the Jews had a treaty with the Givonim) — a far cry from “something that makes the Goyim think less of frum Jews”.
I haven’t learned the Rambam recently, and the concept is complex, but from my memory, this is how I understand it:
The primary definition of Chillul Hashem is to commit an aveira intentionally. If this were done in front of a minyan of Jews (as DH mentions), it would be Chillul Hashem in public. That would be a Chillul Hashem according to all opinions.
Additionally there is a secondary aspect to Chillul Hashem which is when someone acts in a way that decreases Hashem’s Kovod in the world. This is understood as to mean if someone acts in a way that is not befitting to him. He is doing something that is not perfectly right. I believe there is somewhat of a machlokes if this only applies when doing something objectively morally wrong, or even if it is only wrong because people expect more of that individual (i.e. the world holds Jews to a higher standard — perhaps due to antisemitism and they’re looking for problems, perhaps because they instinctively understand we have more obligations then them — either way if your action would be unbefitting to their perception of how you should behave)
Finally there is the general conept of “yehi shem shamayim misahev al yadcha”, that a person has an obligation to make Hashem “loved” through his actions, and that people should say “fortunate is one who teaches his child Torah”. Colloquially this concept has also been lumped under the concepts of Kiddush Hashem/ Chillul Hashem, although I’m not sure it’s technically the same mitzva.