In one of the early episodes (from 2007), there is a dialog between two men and a woman where the concepts of "boyfriend" and "ex-boyfriend" are discussed. The dialog clearly meant to imply the American English connotation of "boyfriend" - that is, a man the woman is in a dating relationship - somewhat serious, probably exclusive (not also dating other people), but not necessarily intimate (certainly not in Korean culture!). Maybe they hold hands occasionally.
Contrast with the concept of a "friend who is male" - just a generic friend, with no implication of a relationship.
OK, when I played that lesson for my native-Korean girlfriend (who is definitely 애인/연인!), she said that namga chingu (남자 친구), while literally meaning "man friend" or "boy friend", did not have the connotation of "boyfriend" that we have in American English. According to her, 남자 친구 is used for any male friend and does not imply any relationship. And therefore the idea of "ex-남자 친구" is somewhat odd, perhaps just meaning a man who you aren't friendly with any more, but not a "breakup" as was clearly implied in the lesson. She thought that 애인 (aein - sweetheart/lover) was maybe more correct, though I'm not sure the intimacy implied with "lover" was what the lesson was trying to portray.
To get a 3rd opinion, we called her 오빠 (oppa - older brother) in Korea and asked him. He agreed with my girlfriend that 남자 친구 does not imply a relationship, just a generic male friend, and that 애인 might be closer.
I don't doubt the knowledge of the hosts, so my theory - which my girlfriend and her 오빠 think is plausible - is that the younger generation has adopted the American connotation of "boyfriend" when using 남자 친구, since the translation is exact. So if a girl says that a guy is her 남자 친구, they are "dating" more or less in the American sense. We are all in our 40s, and her brother's son is too young to ask about these concepts. So maybe we're just a bit out of touch with the evolution of the meaning of 남자 친구?