There is no single Ramayana. There is a Sanskrit epic attributed to Valmiki, and there are a very large number of retellings in other languages, some of which change the story fundamentally, and the differences are not decorative.
Valmiki's text, assembled over centuries and usually placed in the last few centuries BCE and the early centuries CE, presents Rama largely as an exemplary man, the ideal son, husband and king. The explicit identification of him as an avatar of Vishnu is concentrated in the first and last books, which most scholars regard as later additions to a core of books two through six. The earlier layer is a heroic narrative; the divinity is added around it.
Tulsidas, writing the Ramcharitmanas in Awadhi in the sixteenth century, wrote something else: a devotional work in which Rama is God from the first line, and the reader's relationship to him is bhakti. This is the version most north Indians actually know, and it is recited and performed far more than the Sanskrit.
The Kamban Ramayanam in Tamil, the Bengali version of Krittibas, the Jain retellings in which Ravana is a tragic figure rather than a demon and Lakshmana rather than Rama does the killing, the Thai Ramakien, the Javanese and Balinese versions, these are not translations. A. K. Ramanujan's essay counting three hundred Ramayanas made this general point and became controversial in India for it.
Which Rama a person means depends on which telling they grew up inside.