Talking about sex publicly should be embarrassing to all normal girls, the panties picking stuff is most likely a joke since she knows he doesn't like her to go around talking about their sex life, and I don't see the appointment thing has anything to do here, it could be with anyone. Furthermore, if the story was true, the artist could've easily made it more realistic by letting another girl call Akira, not the one who is sitting right there listening to her. The reason he didn't, I think, is to imply that the story is not real.
P/S: Don't get me wrong, I'm not one of those newbies who stay up until 4 in the morning after reading a NTR story, in fact I enjoy them. Also I know this artist has a ton of NTR stories, but this one seems a bit different.
If you paid close attention, the whole thing was made up, since Eri - the girl who called Yuki's boyfriend in the story, is sitting there looking surprised at the end.
P/S : She's from another piece of this artist, tiltled "Good At Pleading" :D