See the entry nr. 3 here: “Accepter pleinement de vivre avec une réalité qui nous touche plus ou moins durement mais dont on n’est pas responsable”

it means indeed she accepts that she likes to eat and does not try to hide it or to fight against it.

Yes, I think you got the meaning right. As for a translation, how about:

Anne loves to eat, and has no problem with that.

Maybe there are better ways to translate ‘gourmand’ : she has a bit of a sweet tooth, she is a gourmet, she likes her food — but I think the focus here was on ‘assumer’.

In the first one, “Anne est très gourmande et elle l’assume” means “Anne is greedy and she doesn’t deny it”.

If we ask her whether she is greedy, she will say “yes” and she is not ashamed of it at all.

I wouldn’t say “proud” which sounds a little too strong.

For you information, it’s very common to say “j’assume” in French when you did something that could be seen as unusual or even shocking, but that you do not deny and do not complain about. Same thing with “il doit assumer” when you talk about someone else, to explain that he got what he deserved and should not complain:

• “Je suis fatigué en ce moment car je travaille à mon projet 80 heures par semaine. J’assume.” = “I’m tired because I work 80 hours
  per week on my personal project.”

• “Tu as crié sur ton chef et il t’a renvoyé. Tu dois assumer.” = “You shouted at your boss and he fired you.”