Currently, I have four NANYA:240P-DDR2-2GB-PC6400. installedinmy Dell OptiPlex 780 Tower Desktop. When I used the advisor tool it said Crucial 8GB Kit (2 x 4GB) DDR3L-1600 UDIMM was compatible with my computer... Does this mean my computer can take both DDR2 & DDR3
Most systems do not support multiple generations of RAM so it is good you noticed the discrepancy.
You can go to the Dell support site and enter your Tag number to get access to a User Manual & Service Guide for your computer as well as information about the original configuration. Usually the Service Guide or other linked documentation will give you some technical information on the memory supported by the system. Just searching in general for an "Optiplex 780" I found this Dell Service Guide which states it uses DDR3 1067MHz RAM (or PC3-8500) which confirms Crucial's recommendation for an Optiplex 780.
I would run the Crucial System Scanner on your computer to see if it comes up with a more appropriate suggestion.
I would also go into your BIOS and verify the model of the computer and the Tag number to make sure someone did not swap motherboards between two cases.