I am another one of those people that just "assume"
It is a one yr old birthday so (hypothetically here) I wouldn't bring my older boys or even possibly DH but yes I would "assume" the invite included at least my younger kids as well. If I did go without the kids I would be quite upset to see other children there knowing that mine weren't invited, it would put my nose out quite a bit.

My DD received and birthday invite that only had her name on it and again I "assumed" I was invited given that she is only 20months. Was that wrong of me?

I guess in part it does come down to the way you were raised. I am of the all or none school and the "more the merrier". Obviously there are times and places where this isnt appropriate eg weddings but for a first birthday I dont think it was wrong of her to assume it was a family invite.