** Adding This:
I read your "so what happened."
I know your baby is a baby. So that is fine.
But later as your baby grows up and gets older like about 1 year old, and your eldest child or younger child is invited to parties, then you would need to toggle the kids and who stays home with them or whether you'd need to take all your kids to a party that the other was invited to etc.
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When there are siblings, no matter how old and there is a party for one of your children to attend a party and you accept... then you have your Husband or someone else, watch your baby.
If you cannot have someone else watch your other child, then you tell the Host of the party. But the party, is for your invited child to attend and to be with his friends. And it is the invited guests of the party, that will receive a goody bag etc. Then, if the party is at some type of venue where food and activity is a part of the party package, which the Hosting family pays for, for the invited guests... then, for the other attendees aside from the invited child or parent... they either can or cannot "expect" that the food there and the activities are including them. Because- the Host "pays" for each invited guest and for the number of food/meals that are required. And they must give the venue a headcount of invited guests only, so that the venue can likewise tell the Host how much their "party package" will cost and know how much food to make for the party. Beyond that, if there are other attendees besides the invited child, then those people pay for their own lunch/food and if they participate in the activities.
If the party is at a person's house... then it is often more flexible... and other people can eat and do whatever activities are at the party... IF there is enough, for all. Invited or not. And if it is okay with the Host.
Then, the other side of the coin is this: as a child gets older... and they get invited to a party, the invited child ALSO needs to have their own time too... meaning, without their sibling. It is THEIR party they got invited too. ie: I have a friend with 3 kids. When one of them gets invited to a party... her Husband will stay home with the other siblings. The Mom goes to the party with her invited child (if the party includes a parent and their child), only. And her reason is this: her child that got invited, has a right to go places too, by themselves to their own parties, without their siblings and so they feel they have their own thing. She explains to the other siblings that this is the other child's special time, with his friends. She does not feel, that ALL the siblings have to go EVERYWHERE together all the time to every single place. She feels, it is important for a sibling to have their own things and their own parties to attend, on their own. Each of her 3 kids, are fine with that. They are all young and in elementary school. The youngest being in Preschool. She feels it is important for each of her kids, to have their own friends and their own things to attend. Even on play dates. She does not bring ALL of her kids to a play date, if ONE of her kids gets invited to a person's home. She says, it is that child's special time.... without all of his siblings. They need separate time, too. Which is important.
I do the same. Unless a parent tells me that ALL my kids are invited.
No matter what, whether it is a party or a play-date... ONLY the invited child (and their parent) attends the party.
It is rude... to attend a party or play-date bringing other people/kids who were not invited nor on the RSVP list, because... the Host cannot possibly be "expected" to provide food and budget for all. A party or play date is for the invited child. Not for a "family." Unless, you speak to the Host and it is okay with the Host.
As an example:
My daughter was once invited to a party at a venue. Dave & Buster's kind of place. The Host, was providing lunch/party favors for the invited kids. It was a drop-off party. No parents. The Host needed a hard head count of who would be attending, because she THEN had to provide the venue with that head count... so that the venue could then provide a table with enough seating and know how many plates to provide and what size cake etc. and how many venue provided favor bags, to make. However, my daughter wanted me at the party. She was younger then and shy. She didn't know any of the other invited kids. So, I spoke to the Host (whom I know), and I explained. She then told me it was fine if I attend with my daughter... but, due to HER budget and costs, she could not "buy" lunch for me, and she had already given the venue a down payment and head count by then. But she said that I was welcomed to come to the party. It was fine... I went and bought my own lunch. And I did not "expect" anything to be paid, for me. The party was for the kids that were invited. I also did not bring, my younger child with me. My Husband was home with my son.
So, there are many things to think about, per a party and when a child is invited to a party.