There are two luni-solar calendars are used in India. Poornima months and Amavasya months. Poornima months months starts with krishna paksha and end with shukla paksha where as in Amavasya month starts with shukla paksha and ends with krishna paksha. There are equal amount of treatise exists for both system. Amavasya is easy for calculation of Panchang, where as poornima becomes more complicated during Adhik month and kshaya maas, (if you look at how they are dividing the months when Adhik month comes, you could end up scratching your head). Amavasya is very easy. Poornima is mostly used in Hindi speaking belt. Non-Hindu speaking belt like gujarati, Marathi, south Indians uses Amavasya month system. However, gujarati's new year starts with Kartik shukla paksha pratham where as Hindi speaking people, and south Indian's new year start with Chaitra shukla paksha pratham

In Poornima Krishna paksha comes first, and then shukla paksha. so when Amavasya months have posh krishna paksha Poornima will have Magha krishna paksha and then will be Magha shukla paksha (common for both), after that Phalguna krishna paksha will come for Poornima months whereas amantha month call it as a Magha krishna paksha.

When I say Magha krishna paksha Chaturdashi for Amavasya it also means Phalguna krishna paksha Chaturdashi for Poornima