LCD Driver Has Own Voltage Boost Converter After All
I took apart an AT&T CL84209 cordless phone system and kept one of the handset's circuit boards attached to its LCD unit. This allowed me to power up the circuit board and probe how its main processor controlled what's shown on the LCD. I then took the LCD off of its board to take a closer look at the related circuitry.

Probing data indicated there was +5.2V on one of the LCD pins, and I had guessed it was supplied by a voltage boost converter somewhere else on the circuit board. I learned what they usually look like from the time I dug into the circuit board for a laptop screen, which included a boost converter to power its white LED backlight. On this handset circuit board, I see at least one assembly of an inductor next to a diode close to a capacitor, all the elements I now associated with a boost coverter.
But when I traced the phone handset circuit board, I didn't find a +5.2V line going to the LCD. I found only a pair of capacitors. One of which connected the two square-wave generating pins (7 and 8) and another capacitor connected the +5.2V line to the +3.3V line. (Pins 5 and 6.) I don't recognize what this arrangement does, but I could replicate it on my breadboard and see what happens.

The analog capture showed what happened: We now see pin 6 sit at 3.3V immediately upon initial power-up and rise up to +5.4V in response to the first I2C configuration message. Finally matching the pattern I saw on the original capture. This is amazing! The LCD driver chip only needs a +3.3V power supply and from that it generates its own +5.4V. There must be a built-in voltage boost converter with inductor and diode (or their functional equivalents) internal to work alongside an external capacitor. I thought I had learned enough about boost converters to recognize them when I see them, now I know I'm wrong. Earlier I had put a capacitor between the boost converter output line and ground, and that didn't do what it did when I put the capacitor between boost output and +3.3V. I clearly have more to learn more about boost converters. But at least now I'm finally driving this LCD properly.