You start at 0 & you classify as hours for the first 24, then days for the first month & then months for the first 12–24 months. The day you are born is your “date of birth” where 1 year later is your “birth day” aka birthday. On the day you are born you haven’t lived any time until after the fact, so you can’t logically be 1 anything until after you’ve lived it.
You are considered a “newborn” from 0–2 months, a “baby” from 2 months-1 year, & a “toddler” from 1–4 years. Don’t know if that’s why the delay in the birthday or not.