Hi L.,
I know your mom says you shouldn't do something for your baby's first birthday. It is true that she will not remember, but you will!! And when she is older, looking at old photographs, she will be looking for pics of her 1st birthday. This is something that you should do, definitely!
Now, the hard part is to make it work for everyone. What about a small get together on Saturday or Sunday after Christmas? Or even 2 small parties, one with family, one with friends?
Or, you can do what I do, with family only, for my son's birthday - he was born on Thanksgiving day, so that's when we celebrate. After our big dinner, we have birthday cake, then do presents. My family is not that big, just my parents and my kids (4), and a family friend that shows up. Talk to your parents, see if that is ok. Family will already be there, they just have to share some cake for your daughter. And birthdays dont have to be celebrated exactly on the day.
I know money is tight, but there are cheap/quick ways to feed everyone. Meatballs in barbecue sause, chicken with sauce, chicken wings, in tin pans with oil burners and salads. (We did this for my daughter's graduation party - she made 2 different chicken dishes, one was hot wing dip.) Is it possible to make a cake or two at your parents house? It is less to carry with you. And, dont forget any leftovers you can get your hands on; like pumpking pie, turkey or any other meat, you can make sandwiches out of.
Is there a friend who will let you use their house, or a community hall/grange hall/fire hall where you can hold a small party, so you dont have to have all of them at your parents?
Invite who you want, which from what I read will be alot of people, if times are as hard for everyone, or if they feel jilted because you cannot attend their functions, some may not show up at all. But, if they are true friends, they will show. (There is only so much you can from your long distance, do not feel badly about it)
I know it will be busy for you, since it is so close to a big holiday and alot of work, but it will be worth it for your daughter and your own peace of mind - you are not letting her big day go unnoticed because of it being so close to the holidays.
As for presents, I don't think you should put not to bring presents, some may feel badly about it. Just put that their presence would be much appreciated and leave it at that.
Good luck