Penny de Valk's Answer:
Your decision to speak to her one-to-one is a good one. You don't need to specifically refer to your cultural differences but to be cognisant of them when you are speaking to her. In fact it is an opportunity for you to demonstrate your sensitivity to these differences without being overt about it.
You will be aware that these 'direct' conversations may seem quite challenging but if done in a conciliatory way using language like "when you...I feel..." as opposed to "you are being rude and dismissive etc" might get to the bottom of what her problem is, recognising that she is unlikely to tell you directly if she has a problem with what you are doing.
Instead try to understand her motives by asking open questions about how you might enable her success, checking in with her about her own career aspirations and what suggestions she has for how things could be improved - it might uncover what she is trying to 'prove'.
Good luck with the meeting and in your career generally - it is a real challenge trying to be an effective manager outside of our cultural milieu but an important one for managers to embark on. You are a pioneer and that is not always an easy role to play!