I watched it four times over the past three years, and I have come to the conclusion that Jae On never lied to her when she asked if he is seeing someone else (because while he's a playboy in the beginning, he doesn't make a secret out of it, and they also never promised to be in a relationship, so he has no reason to lie to her. Besides, it's against his character), and possibly didn't even see anyone else in the entire time he was involved with Nabi (after that first kiss/sex). We are led to believe that he's insincere and he would sleep with other women (because that's what Nabi and everyone else expects) but the two times it is implied (Seol Ah and that older woman whose brother punches him), it turns out to be not true.
I think that's why Sol is such an important 'witness' to the relationship. Contrary to everyone else, she's very supportive and tells Na Bi that she thinks Jae On is much better than her ex and they make a cute couple. Why? Because she's the only one who doesn't care for gossip, so she only reacts to what she sees, and Jae On is actually pretty sweet to Na Bi (especially considering they're not even dating).
And the couple of times he got a call and actually left Na Bi, it seemed to be always Seol Ah. Usually it turned out to be a visit to the hospital since she's in bad health. I don't think they had anything going on at all after she came back to Korea this time.
I think the only time it is implied he suggest sleeping with someone is indeed the woman at the bar, after NaBi ends it for good and vanishes into thin air, but even then we cannot be sure.
All in all, I think a lot of viewers paint Jae On very black, when he never lied in the first place.