I've always thought that saying you're one year old after one year of birth, two years old until 2 years after birth, etc. is fine. I do think we need to acknowledge as a society though that it's only counting from birth, it's not counting from when you were first alive.
I've always thought there was a problem in a way with counting from birth for premature children. Their actual age is younger than their birthday would indicate, and so they don't necessarily reach "age milestones" at the same time as other kids. This isn't necessarily because of problems in the womb, but it could just be that they are actually a month or two younger than a child that should have reached a certain milestone at their "age".