You didn't understand my analogy at all. Both fires still use wood while burning. Only the fire that burns brighter, needs more of it.
So if a display needs to make a part of it brighter, it will use more battery power than it uses while the display has less bright colors to display.
The problem with your analogy is that there are two fires, one per pixel. An LCD does use a tiny bit of power per pixel, but most of the power is used by the backlight. In a phone LCD there is almost always a single backlight unit for the whole display. There is not a separate light per pixel.
You can't 'brighten' one pixel and 'darken' another by sending more or less power to the backlight.
You can only let the light through, or block the light. That's how LCDs with backlights work. Displaying a black pixel doesn't use less power because the backlight is still on. This is why an LCD monitor that is 'on' but displaying a black screen looks different to one that is truly turned off.
If an LCD has a backlight that at full brighness uses one watt of power, that backlight will continue to use one watt of power whether or not you are letting all of that light through to your eyes (white screen) or blocking out some of that light before it gets to your eyes (black pixels).
The only way to significantly reduce the power output of a backlit phone LCD is to dim the entire backlight by turning down your phone's brightness setting or having it auto-dim.
Edit: I made a diagram, maybe this will help:
Turning the pixels on or off has no effect on how much power the backlight uses.