Birth Certificate Typos?

My last name was spelled with a typo, and my moms last name was also spelled incorrectly on my birth certificate. I called the new york city of vital records and they said that if im under 18 all i would need is my mom to bring a copy of her birth certificate and mines with a copy . I just wanted to know if i turn 18 and i don't have government id? What else can i use to for valid id? Would i need a notary to detail the rel;ationship between me and my mother and can my mom do it on my behalf upon turning 18?