In reality you're 9 months old when you're born because you began living by the time you started to develop in the mothers womb.
And in some cultures people do count the 9 months as the age of the baby when they're born but in the USA we count the babies and our ages from the minute we're born and come out into the world alive and healthy.
So our birthday is celebrated on the day we were born and we're considered a day old when the day we born passes.
Birth is also seen as the moment in time that the baby starts to use its own individually breathing body, and is not a part of the mother’s body anymore.