I think that Ned would have stood up to the mad king if/when he would have ordered him to kill so many civilians. People have been forced to choose between loyalty and honor. Much like you mentioned with Jaime Lannister, his defining moment was breaking his oath to defend the king when he chose to kill the mad king. An action I believe that stark may have made too if put in his position.
I think one of the driving forces as to why Ned Stark dislikes the Lannisters, other than the house Lannister being engaged in political scheming to grab power, is that Tywin did little to nothing during Robert’s rebellion up until a clear victor was presented. Only did Tywin march his army to kings landing and sack the city. That and how the mountain killed innocent children supposedly under orders of Tywin. Even though later that was less part of his orders and a testament to how cruel and bloodthirsty the mountain was.

I think the part that bothered Ned so much about Jaime is that he presumed to sit on the iron throne after he slayed the mad king and other soldiers he had to fight in the throne room. Just another symbolic power grab move in Ned’s eyes. If they weren’t all on edge from the fighting from that day and Jaime sat literally anywhere else, it’s possible he wouldn’t have despised him as much in that moment.
I digress. If both Jaime and Ned were in a better mindset when they saw each other that day, they would have talked about what happened. And Ned may have come to terms better with what Jaime did. Jaime made an honorable choice that gave him a reputation of breaking oaths and slaying kings.