N.
Wow! With 5 kids, you've got a lot of this kind of stuff to look forward to, don't you?? :-)LOL
I think what you did was fine. When my oldest son was little, his playmates were on the same street or his best friends were on his sports teams so we knew the parents and just arranged for "playdates" in person after games and things like that. When my daughter, now 9, first started Kindergarten, you could have knocked me over with a feather when her classmate's mom called me up to arrange a play date at their house one day. I felt ridiculous because I had no idea how to respond. I mean, in my head I was thinking "Is there an assumption that I go too?", "Who are these people?", "If they invite her over, do I have to invite them over too?", etc. Fortunately the other girl's mom was a pro at all that and pretty much walked me through it. She said I was invited to come as well so we could meet and chat while the girls played. She told me she always invites the parent the first time as that's what she would expect if someone called to arrange a playdate with her daughter.
With birthdays, we always wait for the official invite 'cause kids tend to have ideas in their head about what's going on and it's rarely what is really the plan. In your case, however, I don't think what you did was bad. The only "bad" thing would be if the other parent was not planning on inviting your son and then feels obligated to because of your note. As a parent, I personally would not feel that way and would politely call or send a note back stating that my daughter was mistaken about her birthday party plans and I would apologize for the misunderstanding.
That said, be prepared for some highly uncomfortable moments, even when parents are the people doing the inviting. I was at my daughter's girl scout meeting and one of her friend's moms came up to me and told me about her daughter's birthday slumber party and that my daughter was invited. A couple weeks later, my daughter came home crying saying that the other girl told her she was not invited because the mom said that when they get together, my daughter is too hyper. I personally didn't mind that my daughter was no longer invited because the other girl is NOT the best of influences, but I know it broke my daughter's heart, and the fact that the mom didn't even call to explain to me about the situation...well, let's just say that we don't have any dealings with her or her daughter anymore if we can help it.
Anyway, I digress. My point is that when dealing with other parents regarding parties and playdates, you just never know if you're going to experience someone very nice and helpful like the first mom who invited my daughter over for a playdate with her daughter, or if you're going to have dealings with other parents who aren't quite up to par on proper social etiquette.
I say just do what YOU think is right and you'll learn your lessons along the way. You've got 5 kids to work with so by the time your 8 week old is in Kindergarten, you'll probably be able to write a book about kids, their parents, and every social situation imaginable! :-D
Good luck!