So long as he doesn't disrespect her and will still serve her a drink, she'll allow him to believe whatever he wants.
Cora likes everyone :D.
I'm sure most who had been at New Haven a while have seen what she can do on one or two occasions, so Logan is well informed :).
I can see Verity and Ben having a bit of a mutual respect. They've both had temper issues and he's got his big heart and she's got her passion. I can see them as friendly if not friends.