I'm surprised at how many people have said that they would just assume that their children/partners were invited to an event if it was only their name stated on the invite. If it was only my name on the invite I'd assume only I was invited and I'd then decide if I wanted to go without my family, I wouldn't end the friendship over it especially considering it's a uni friend. If it were family or a really close friend who knew my dd really well and they had invited other children then I'd be hurt by it and probably not go. I think it's an awkward situation now that she has assumed her whole family is invited. Had she questioned just her name on the invite it would have given you the opportunity to say sorry we are trying to keep it small, I'd love you there because of your support during my pregnancy etc. I don't think you're being rude. A first birthday is so special and I understand you only wanting ppl there who have had a part in your and your dd journey so far. I hope she has a wonderful first birthday
Hugs!