Regarding Date of Birth mismatch (in month column) in documents

Sir/madam, I recently graduated with masters degree in India and I am selected for PhD abroad. I am struggling with the following situation. I had a passport when I was 4-5 years old (validity 2000-2005). I lost my original birth certificate very early during primary school, and the passport was also never used after age of 5 (year 2001). The DoB mentioned in my old passport (linked with original birth certificate) is 19-March-1996. I had to change multiple cities and schools during school education. Error in month column of DoB happened very early during primary education. I passed class 10th in year 2011. The month in DoB mentioned in CBSE highschool marksheet was June. I used this DoB throughout my education, and in all documents e.g. during making aadhar, bank passbook, PAN, DL, etc. I got the opportunity for admission in PhD abroad, recently. For this purpose of higher studies and career, when I went to make passport, Passport office could reissue my passport only with reference to previous passport and original birth certificate registered with them. They guided me to correct DoB in other documents. I was able to correct DoB in Aadhaar. Also thankfully I could get relevant details of my original birth certificate and based on those details I now have obtained reissued copy of original Birth certificate. At present my reissued passport (July 2024), reissued birth certificate (August 2024) and updated Adhaar (August 2024) have same DoB, whereas in 10th CBSE marksheet and other documents like PAN, DL, etc. the DoB is incorrect. I wanted advice what is the appropriate procedure so that this mismatch does not become obstacle in my PhD abroad and future career. CBSE authorities have asked me to apply through school with supporting documents. School is not able to guide and support in this matter. My 10th education was done in Uttarakhand.