Here’s some data, from Gyula Moravcsik’s Byzantinoturcica, a dictionary of all Turkic names and words that ended up in Byzantine Greek. The names are in roughly chronological order.
Osman is named as:
- Atman (George Pachymeres, Nicephorus Gregoras)
- Atouman Atoumanos Atoumanes (Notitiae Chronicae, Chronicon Turcorum)
- Otmanos Otmanes (Hierax, Chronica Minora)
- Otoumanos (Chalcocondyles)
- Othmanos Othman Otthmanos Otthmanes Otthamanes
From Turkish Atman, Ataman, Azman, Tuman? ~ Arabic Otmān.
So Moravcsik accepts that the name originated as Atman, with abundant bibliography, and the very earliest mentions are as Atman. The versions with an initial O- and with a -th- are 15th and 16th century.