Two Adars, one birthday
A leap year adds a second Adar. The date you were born and the date you celebrate answer two different questions.

Keep the original month
A Hebrew leap year contains Adar I and Adar II. If your birth date converts to one of these months, keep that distinction in the record. “Adar” alone can hide information you will need when looking at future anniversaries.
Conversion asks which Hebrew date corresponded to your actual birth. It should preserve that date even if a later celebration follows another convention. Start with the original record, then ask the anniversary question separately.
Celebration customs differ
Hebcal’s Ashkenazic birthday algorithm places an ordinary-year Adar birthday in Adar II when the anniversary falls in a leap year. Chabad describes celebrating in both Adars for someone born in an ordinary year. These approaches illustrate why the event and community matter.
A birthday, a coming-of-age observance and another anniversary should not be assumed to follow identical rules. When an observance has religious significance, ask about the specific event and the custom your community follows.
A twelvefold text in a thirteen-month year
The Gra version of Sefer Yetzirah presents a twelvefold pattern and names Adar. It does not supply a separate modern personality scheme for thirteen months. Mazalogy groups both Adars with Pisces and Qof as a stated editorial convention.
The chart can therefore preserve your actual Adar month while sharing a symbolic theme. Let that distinction remain visible. A calendar fact deserves precision; a reflection deserves the freedom to become a useful question.