How can I identify a salvaged LCD? 21 pins
The different sections of the LCD light up when the LCD is connected
to power and you ground the different pins.

The number of pins and segments on your display suggest that it is a multiplexed display similar to this:-

You can determine the pinout of your LCD by noting which segment activates when you apply a voltage between any two pins. After recording the response to all possible pin combinations you should be able to produce a segment layout map similar to the one above.
Each LCD segment is 'lit' (changes polarization) when a high enough voltage is applied between its segment 'row' (number) and common 'column' (letter). To avoid lighting other unwanted segments at the same time, multiple voltage levels are applied. Only those segments with a voltage difference exceeding the LCD's threshold voltage will 'light'. To avoid damaging the display the average voltage across each segment must equate to zero (ie. AC voltage only applied), so all the voltages must be periodically inverted.
Shown below are examples of 3 and 4 level drive voltage waveforms. Only the periods shown in orange have enough voltage difference to 'light' the segment.

To produce the multilevel voltages, resistor dividers can be applied to the MCU I/O pins, like this:-

The 3 levels are produced by setting the I/O pin high, low, or open circuit (when the resistors set the voltage to 1/2 Vcc). Your driver board has several resistors on it that might perform this function.
All the images above come from an article which describes how to drive this type of display from an AVR MCU.
Awawa's Lab: Direct LCD drive by AVR GPIO