Did Sai Baba ever refer to himself as God, and how did he get his name?

According to the source material, Sai Baba displayed great humility and never claimed divinity for himself. Chapter 0 explicitly states that he never uttered "Anal Haq" (I am God), instead referring to himself as a "banda" (a servant of God) and frequently saying "Yaade Haq" (I remember/meditate on God). He acquired the name "Sai Baba" upon his arrival in Shirdi. As chapter 0 recounts, when he entered the Khandoba temple, a resident named Mhalasapati welcomed him with the words "Aavo Sai Baba" (Come, Sai Baba). He carried this name for the rest of his life, even confirming in a commission, "They call me Sai Baba."