You've already had some great responses here, but i'll toss my 2c in anyway

1) The female pelvis is NOT fully formed at 16, it often isn't fully formed until 20 or 21! Even once it is fully formed, unless a woman is in labour and in the best possible birthing position (deep squat or hands and knees) it's impossible to see how much room she has for a baby to pass through. If the x-ray had been taken while you were IN labour and the baby's head was as descended as it could get, i can imagine that one x-ray might be able to reveal if/why you couldn't birth that specific baby in the positions you were both in. But that's it. It can't tell you anything unless you're in labour, and if you are in labour it can only tell you something about THAT labour, not any potential future ones.

2) there ARE pelvises which make birth more drawn-out and posterior presentation of the babe more likely (grab a pink kit for more info on that, it's related to the angle of the front pubic arch) BUT and it's a huge BUT, those women can ABSOLUTELY give birth vaginally. Their labours might not look "textbook" but they are perfectly normal, healthy effective labours and they get the baby out just fine in the end.

3) looking at family history can be helpful, but fear is contagious, more contagious than anything else that will affect your ability to birth. If the women of your family couldn't give birth to their babies you'd have died out about 150 years ago when sections weren't available/safe. It is VERY possible that your family have the shape of pelvis mentioned in the pink kit which can slightly more often mean a longer labour and posterior presentation (and as someone above said, it's not something you can see by looking at your bodyshape), but it is the Ob panicking and prescribing a section which prevented vaginal birth, NOT the pelvis. On the whole 13 hours isn't at all long for a first labour. Up to 24 hours is totally normal.

Obviously i'm not a birth psychic, and some women DO need a section, but less than half of the women who are currently having them ACTUALLY need them. Without knowing about your mum's first and your sis's only labours it's hard to say for sure, but i'd be willing to bet intervention made some difference in those situations. Were they moving freely and upright during labour, or were they made to lie on the bed most of the time? Did they manage to avoid epidural anaesthesia and stay as drug free as possible? Were either of them given induction drugs or was their labour un-augmented and totally natural? There are SO many factors in birthing a baby, and the shape of the bones is realy, truly, very rarely a major one. Look at it rationally - your guts would easily fit through your pelvis, but they don't fall out because your pelvic floor holds them in place, that same pelvic floor holds the weight of your baby as it grows, bounces, wriggles and squirms and then one fine day it doesn't anymore, instead it completely reverses its normal function in order to let your baby out, and right away afterwards it reverts to its normal form and even though a giant baby fit through there, your guts STILL don't fall out! - the female anatomy is a many-wondrous thing hun, the more i learn about it the more astounded i am by it.

If i were you i'd perhaps seek a different opinion from a MIDWIFE. They see every size and shape of woman and they are very in touch with what is normal and possible, and will be able to offer you a realistic viewpoint and give you more confidence in making your decision. You might also want to consider a different Ob if you feel yours doesn't listen to your concerns - however you birth you need to be able to trust your care-givers and putting yourself and your baby into the hands of someone you already find to be ignorant of your concerns and needs.

Best of luck

Bx