I feel I have a working understanding of the I2C control messages necessary for controlling an LCD salvaged from an AT&T CL84209 landline phone system. But that is only part of the puzzle, because this LCD had input/output pins beyond I2C clock and data lines: there were four more pins to figure out. Represented by channel 4-7 in this logic analyzer trace:

Upon system power-up, one started at 3.3V and the others at 0V. Roughly two seconds after power-up, they come alive with two of them going up to 5.2V and the other two generating ~8kHz square waves. One between 0V and 3.3V, the other between 3.3V and 5.2V. I had assumed I would have to build circuitry to mimic these voltage values, but maybe I wouldn't.

When zoomed in to when these pins became active, I could see the analog activity was immediately preceded by activity on the I2C bus. Comparing the timestamp against the list of decoded I2C messages, I found the I2C message visible on this graph is actually the very first set of messages sent to the LCD at address 0x3E.

This implies the LCD is not just a passive receiver of these signals, but an active participant who started something in response to the first initialization command sent over I2C bus. What we see here is some combination of functionality onboard the embedded chip, and some contribution from the circuit board. But I have no guesses at the division of labor between them. I assume the circuit board must contribute something to the process, otherwise the embedded controller could have just kept things internal and not incur the manufacturing cost of bringing those pins out.

I looked to the Sanyo LC75853N LCD controller datasheet I previously referenced for a different device to see if it had a counterpart to these pins. I note the OSC pin that required an external resistor and capacitor, and the VDD1 and VDD2 pins that required external voltage supply. None of these explanations fit the two pins that rose to 5.2V. The square wave between 0V and 3.3V might be an external oscillator, but I don't know what that means for the other square wave between 3.3V and 5.2V.

On a quest for more data, I am going to try wiring these two LCDs in parallel again. The previous time was a risky experiment done in ignorance armed only with the fact I had very little to lose. This time I'm a little better informed which I hoped should translate to lower risk on the second try.