It's definitely a birthday, hence the name. It's the day you turn 0.

It's a definition that could add 1 to everyone's birthday celebrations. A baby born on the 15th of June 2024 could be celebrating its 2nd birth day in a few days. Or well the family would. They'd celebrate the birth, and why not count that as a "birthday celebration".. Ha.

Curiously I guess most of us celebrate the number of years completed.

They would be celebrating their first anniversary.

This pops a lot with annual celebrations. I attended a commencement for my niece several years ago, and a speaker referred to it as the 100th annual commencement while a brochure referred to it as the 100th anniversary. Pick one.

You wouldn't remember the party anyway

Same with 12 months after birth