I don't think the question is so much "do I have the right to invite whichever kids I want to" - of course the OP has the right to invite anyone she wants, and yes, only invitees should attend. Siblings who are not invited should NOT attend no matter what the age.
The problem here is that she is really asking "Will my family and friends be offended that I put a note in each invitation that calls out which kids I will pay for and which ones I won't." This is an issue BECAUSE the baby is one year old and doesn't really have friends yet who are not part of a larger family that are likely also considered "friends" of the mother.
So, my thoughts based on what I would do/think:
1. If my child attends daycare with the child in a room specifically for ONE year olds, and none of my other children are in that room , I would feel fine inviting just the one year old. Often with smaller daycares, however, there is a baby/toddler room that can range in age from newborn to 2 1/2 year old, which makes it very feasable that a one year old and a 2 1/2 year old sibling will be in the same room, I would feel obligated to invite both children, but not any older siblings - SO LONG AS this was a daycare/friends ONLY party and I was not really friends with the family outside of daycare. I would not put a note in the invite calling out which kids I would pay for. It seems that it would be understood based on the nature of the relationship.
2. If it is a home daycare and all ages are together and someone has a 1 year old, 3 year old and 5 year old in the daycare with my 1 year old plus maybe even a 9 year old sibling as well, I would invite the whole family. To not invite everyone would be the same to me as a school age kid bringing invites to school and handing them out to a select few and not everyone (which, by the way, is against the rules in every school my kids have attended!). Again, no note neccessary.
3. Family should be invited as a family unit and all kids in the family should be paid for by the host. under no circumstance should a note be put in the invite, nor should any family member be expected to pay for their older kids (which I am assuming would be your neices/nephews?).
4. Non-daycare family friends should be invited as a family unit and all kids in the family should be paid for no matter the age. Again note in the invite.
Like I said in my earlier post, if I received a note in an invite saying that only one of my kids would be paid for, I would decline the invite not because I would want to bring an uninvited sibling, but because I would find the note itself rude.