In the sloka,
‘Raamaaya Raamabhadraaya Raamachandraaya vedhase
Raghunaathaya Naathaya Seetayaah pataye namaha’,
Who called Rama by these names?
The Ramaraksha stotra was recited by Budha Kaushiki Rishi (Valmiki). No one called Rama like That, but Valmiki praises and sings about Rama like this. Now this is a famous shloka and everyone recites it.
Dasharatha – Rama
Kousalya- ramabhadra
Kaikeyi- RamaChandra
Vashishta- Vedase
Rishi- Raghunatha
Sita- natha
Maharaj Janaka- seetapathi
The verse is 27th shloka in Ramraksha Stotra.
The Ramraksha is written by Rushi Bhudhkaushik as Bhagwan Shiv came into his dream and asked to compose so.
The names taken in the shloka are:
Ram – the name of Lord itself.
Rambhadra – Ram + Bhadra (bhadra = auspicious, well doer)
Ramchandra- Brahma, One Who Knows Everything, Brahmaswarup, UniversalRaghunath – Nath (leader/God/One above all) of Raghu vansh/lineageNath – God, Leader, Swami, our Guardian
Sitapati – Husband (pati) of Mata Sita (Avatar of Mata Lakshmi)