What was the biblical name of the religion or faith Avraham belonged to? the word he used to distinguish his faith from other, pagan belief systems?
There’s no indication that Avraham’s belief system had a name. In fact, there is no word used in Tanach for any belief system so far as I know. (Or, for that matter, for the concepts of “belief system” or “philosophy” or “religion”.) Tanach mentions the worship of this pagan god or that pagan pantheon, and contrasts them with the worship of the One God of Israel.
From Jewish tradition associated with Avraham’s hospitality, we can guess at the core of what he himself taught: there is one God, God the All-Sufficing (El Shaddai), to Whom we owe gratitude for everything we have.
But I repeat, there is no word associated with this philosophy or religion found in Tanach.