Lorelai told Sookie that she and Luke broke up after she slept with Chris, her sleeping with Chris was confirmation that her relationship with Luke had ended, and this means that she slept with Chris to prevent any attempt from Luke, which means that Luke did not consider it a breakup and that Lorelai cheated on him with the man who was insecure about him to push Luke to finally separated from her.

As I said, this was not a clear breakup except in Lorelai's mind, so she slept with Chris to prevent Luke from making any attempts, and as I said, if her breakup was final, she would not have needed confirmation of her breakup with him.

Yes, Lorelai's relationship didn't end well with Luke, because she didn't even respect him as a friend, unlike Luke who despite everything Lorelai did, was there for her and her family as usual (7.14), even though he owed her nothing.