R.J.
You know... there's SHY/private/etc. and then there's "Aloof".
It can be difficult to tell the difference, but here's a good rule of thumb:
Aloof kids (like 1 of them) are common in big rowdy families, because it's a way of getting attention. By only answering 1 word, she's making other people stop what they're doing and pry the info out of her. It's how she 'competes'. Others might go on and on drowning other people out (hogging the air waves). Others might interrupt or speak over each other. Others might be early or late to every event. Others might early bird it or night owl it. Some go for 'excellence'. Others go for the 'bad boy' vibe. There are DOZENS of ways that those from large families carve out their "spot" in other people's attention AND their private time. Aloofness is ONE of them.
If she's happy & outgoing & has no issue sharing with other people (friends, teachers, etc.), then she's not shy. She's getting her attention from her parents and siblings by making them pull teeth.
If she's shy or introverted (two VERY different things), she's not manipulating others into paying attention to her by holding back and making them come to her.
Shy extrovert (most common): she'll be WANTING to speak up, but doesn't. If she's an shy extrovert, she'll be hanging around the edges of things, wanting to join in but not unless someone brings her into the mix... the classic 'wallflower' is a shy extrovert. A shy introvert will stay home, a shy extrovert will go out, but be on the outside looking in. This is the kind that does best in a large group with a "wingman" to metaphorically hold her hand / keep bringing her into the conversation. She won't join in on her own, although she wants to. Not a good 1:1 type person, better in a group of people talking.
Shy introvert: She WON'T be wanting to speak up, and is perfectly / most happy by herself or with a very small group / 1:1. This is the kind you can usually have GREAT 1:1 time with, but they need a lot of privacy. If they don't get their alone time, then they CAN'T actually do 1:1 with someone, because it's too exhausting to them.
((Doesn't sound like she's an Outgoing Introvert: (that's me) has NO problem talking to people, joining in, starting things, etc. BUT they need a lot of alone time / privacy. When they HAVE to be around other people; school/ family/ etc., then their "spare" time the usually spend by themselves or with 1 other person. Again, a good 1:1 type person).
From what you describe, I would think she's either aloof, or a shy extrovert.
Aloof, there's not really much you can do about, since it's a choice.
Shy extrovert... ditch the 1:1 for now... and think "spa days" & "girls nights". She'll feel most comfortable talking if there are a group of people talking AND someone is constantly 'inviting' her into the conversation (What do you think, has that ever happened to you, ever known anyone who... etc.). Ideally, ask them 2nd or 3rd or the shyness will make them hold back.