Capacitor C2 along with resistor R10 provide a means of resetting the LCD in hardware. When switched on, the voltage on the LCD Reset pin remains at zero for a short time as the capacitor charges via the resistor, this is effectively a "low" pulse rising to "high" and the LCD, Reset is active low. That's the theory but for me it didn't work no matter what values I used for R10 and C2 - the LCD just would not fire into action - I even tried a second LCD and had tested them both separately and successfully with "standard" wiring and an Arduino running a basic "Hello World" sketch. (By "standard" I mean according to the demonstration sketches which use digital pin 6 as the reset pin).
Saying all that, when powered by a 12V/2.1A mains adapter, the LCD came on unfailingly on the second switch on. That is, "on-off-on" with the power and a brief wait between 'off' and the second 'on', the LCD burst into action every time ... but not with batteries even though I could operate the motors using the keypad. I don't understand this at all.
After a lot of testing, checking of wiring and modifying values for R10 and C2 in order to try out different reset timings, I came to the conclusion that both of my LCDs needed a "by the book" reset - by software toggling the Reset pin at the right time and duration as provided by the PCD8544 library module as modified for this project by Sergey.
Solution - I modified the code as described in the Troubleshooting section, "Could be an issue with hardware (RC) reset". This takes away the battery monitoring function but as there are no spare I/O channels on the Arduino, I didn't really have any alternative. (I'm not intending to use batteries very much either, if at all).
Job done and I hope this helps.
Alan