#"What is the most evident is that her birthday is January 6th in a lunar year(not in the Gregorian calendar)." Rubbish! The date "January 6th" only exists in the Gregorian calendar (and its predecessor the Julian calendar). If you want to argue for "January 6th in a lunar year", please present WP:RS evidence of a lunar calendar that has "January 6th" as a date within it. 
 #en.wikipedfia.org is an English language wikipedia. As such, its default calendar is the calendar prevalent among English speakers, the Gregorian calendar. This can be seen by the fact that WP:DATED deals entirely with Gregorian calendar dates. It is therefore the calendar that all DOBs should be presented in. 
 ''''(P)