This is apart from the discussion on the correctness of सीताया: पतये नम: .
I have heard in a kathaakalakshepam by the well known Anantharama Dikshitar the explanation of why these different names are used denoting one person i.e. Srirama.
रामाय रामभद्राय रामचन्द्राय वॆधसे /
रघुनाथाय नाथाय सीताया: पतये नम: //
रामाय – his mother Kausalya calls him Rama.
रामभद्राय – other members of his family call him Ramabhadra.
रामचन्द्राय – his friends call him Ramachandra.
वेधसे – Vasishtha calls him as Vedhas knowing him as the incarnation of Vishnu.
रघुनाथाय – People of Ayodhya who remember Raghu calls him Raghunatha.
नाथाय – Sita calls him Natha.
सीताया: पतये – People in Mithila, Sita’s birth place, call him as the husband of Sita.
नम: – applicable to all the names.