She's selectively forgotten that part or simply thinks of that girl as another person.
This is not uncommon, nor is it necessarily pathological.
When she goes back to that country she tends to dissociate her other self, and blank that episode out and reinvent what she was doing.
The best you've got here is garden variety repression, unless you are saying she is in a fugue state. If she's in a dissociative fugue state, she'll travel to county X for a certain period of time, and then have no memory of it whatsoever (e.g., she "wakes up" in a hotel in Mexico having no idea how she got there or what she did while she was there). Keep in mind, she doesn't just choose not to think about it -- she'll have NO MEMORY of it. At all. Your description doesn't sound like a fugue state, given that you've described her as "selectively forgetting" and as "thinking of that girl as another person", and most importantly, it sounds like she has chosen to visit country X (didn't just wake up there having no clue how that occurred).
I don't think DID is the right diagnosis since it has to be a bigger trauma if I understand right, but it's more of a coping mechanism for the stress of being polarized between two countries.
You're correct that what you've described doesn't even remotely approach D.I.D. You're also correct in describing this as a coping mechanism. Coping mechanisms are not necessarily pathological.
If you want to give this woman a psychiatric condition, you're going to need to include some ways in which this actively interferes with her day-to-day functioning. Without it, all you've got is a person who actively chooses to repress a piece of her past and behaves differently when she's in different situations/countries.
I've looked up types of therapy. What kind of therapy would be good for her? Cognitive, etc kinds.
Again, unless this is problematic for this woman, there is no reason for her to be in therapy. What would prompt her to seek treatment in the first place? If you can answer that question more specifically (i.e., for what reason did she seek treatment, what symptoms is she presenting), you'll get a better answer as far as potential diagnoses. Is she waking up far from home, having no memory how she got there or what she did? Is she depressed about her situation to the point that she's missing work, crying uncontrollably, and losing a lot of weight? Is she so anxious that she has begun literally pulling her hair out?
What has brought her to the point that she is seeking professional intervention?
Craig has hit the nail on the head here. Unless you've left out a big piece of the clinical picture, what you've described is not pathological. Unless this interferes with your character's day-to-day life in a significant way, she's unlikely to meet the diagnostic criteria for any psychological condition, including dissociative disorders.
~STS~