What was the specific process Sai Baba followed when he used water to light the lamps? Did he perform some kind of ritual?

Yes, Chapter 5 describes a specific, deliberate process. After being denied oil, Baba didn't simply pour water into the lamps. First, he took his tin pot which contained a minuscule amount of oil. He poured water into this pot, mixed it with the tiny bit of oil, and then drank the mixture himself, making it an offering to Brahman. Only after performing this act did he take plain water, pour it into the lamps to soak the dry wicks, and then light them. The lamps subsequently burned all night, suggesting this ritualistic act was central to the miracle.