Virinchi Also for birthday (Jayanti), humans and cosmic gods are to be considered differently. For human born like Rama, Krishna, Hanuman etc. Nakshatra is important, not tithi. Chaitra month, Punarvasu nakshatra day for Rama
Luckily tomorrow its punarvasu nakshatra and navami tithi until almost same time. I think Rama Jayanti should be the name of festival and it is difficult for common people to remember when punarvasu nakshatra comes, but its easy to remember 9th day after amavasya, so Rama Navami became popular.
If next year punarvasu and titthi are on seperate days, then conflict arises but people blindly got used to one method