This is all pure headcanon, but: I personally think that due to their size and complexity, iterators don't exactly "wake up" one day, I think it's more of a gradual process as their different components are being constructed and brought online. I imagine that the construction of an iterator is a very long process, taking years or even decades. Because of this I don't think they have "birthdays" so much as "birth years"; I'm not sure where the line is drawn for them to be considered "fully conscious".
In some of her dialogue Moon describes a process of "priming", I think iterators are fed information gradually as they are slowly brought online, and I think the puppet is one of the last components to be installed. So whether their true "activation day" is the day their puppet is installed, or the day their priming is complete, or maybe the day they are unveiled to the public, is unclear. It probably takes a while for an iterator to learn to use their puppet to communicate; I doubt they are fluent in using it right away.
Given the social importance placed on an iterator's puppet, I think the day their puppet is installed would be the closest thing to a "birthday", even if they were technically fully conscious for long before that. It's more of a symbolic gesture than anything else, to "humanize" them to the general public. I'm not sure if the iterator themself would care about it very much, I guess it depends on which one you ask.