
....i dont understand what happened after the wine glass...

My, my, I think my political science degree will put into action trololol. Anyways, she didn't have to apologize when she also spilled the wine to the prince, because in the first place, the prince also spilled a glass of wine to her. Even though you think that it is rude, but actually it's no. Look at how she gracefully laughs it of and said "Now we're even" without any ill intentions. Plus, spilling the wine to the prince also means that "You don't need to apologize and I don't need any favors." So basically by doing that you are stating that "By talking to you, I don't have any intentions of using you. But rather, I wanted to talk to you." And I also must not forget to mention that because of this, she created an atmosphere in which they are in equal footing with each other.
remember that i mentioned equal footing with the prince? That made the Duke (Father of that bitch) To say that her daughter is out of league with the 31st consort, because she knows how to act, she is "friends" with the prince and she is witty.

In fact she went a small step further: she showed the Duke that by demanding she apologise he was actually HURTING the country's rep. Doing so would show weakness, since as you say she had no reason to apologise. That is where the "hidden strength" comment comes in. She's chastising him for implying their country is weak.

A what?

If you're still curious, the translator's note explains it at the bottom of this page:
http://www.mangago.zone/read-manga/neko_neko_baby/uu/iur_chapter-2/pg-30/
Gee why cant i get him to date me? Oh maybe its the fifty girls per week that i have...nah
Exactly. He should've quit dating other people when he realised his feelings and decided to act on it