Thank you for posting the solution to this complicated problem.
I have a similar case – all my documents (school, university, and passport) carry the wrong date of birth. I am applying for J1 home residency waiver, after which I will move to US on H1B. I wish to apply for GC in a year or two, for which I will need my birth certificate.
I am concerned that if I correct my date of birth in passport before applying for H1B, then I will have to change my date of birth in all my school and degree certificates, which is probably impossible.
Should I correct my date of birth in passport (based on birth certificate) only before applying for GC, and therefore, avoid the work and hassle of correcting my date of birth in my school and degrees.
Thank you for all the suggestions.