So, what is 1602, and how do I ascertain compatabilty of an HD44780 LCD with this 1602 standard?
Some form of the term 1602 shows up as in part number of many LCDs that have 2 rows of 16 characters. It also shows up in the part number of serial adapters originally designed for those LCD modules as well.
The exact same controller, an HD44780 or equivalent, is used for most of the parallel character mode LCd modules that are currently available regardless of their configuration, 16x2, 16x1, 8x2, 40x2, 16x4 etc. Furthermore there is absolutely no way to inform the HD44780 of which type of device it is driving. The only difference between driving one LCD and driving another concerns the relationship between the memory address in the controller and the location of the corresponding character on the screen. For more than you need to know about this relationship you can follow the LCD Addressing link at http://web.alfredstate.edu/weimandn.
The bottom line is that any serial controller that is physically able to control one parallel character mode LCD of any configuration should be able to control any other parallel character mode LCD. You may have to tinker with the library to get the addressing to work the way you expect.
I have made no mention of the backlight because that is a whole different bucket of worms. There are no standards for the backlight and all bets are off concerning the compatibility there.
Don