is there any difference at all between the two methods to power the board, assuming that I will be connecting a 12V inverter to power the EL wires?
Strain relief is not going to be an issue, because I will pack everything in a box.
I would be supplying power for the Arduino circuit from a 5V USB outlet, using a cable that only has + and GDN wires, so what plug should I put on other end, i.e. USB-to-header-pin (FTDI), or USB-to-JST-PH (battery)?
It seems to me that the FTDI alternative is more versatile, because of the restriction regarding using the 3V inverter powered from the board itself and a battery, e.g. when developing.
Are you wanting plug in the FTDI on the FTDI header or use a coupling cable to connect the FTDI to the Battery JST? I would just use regular FTDI header pins as originally designed. This will give you the option to modify firmware or whatever else in the future.
There’s not really any difference between using FTDI or battery for power, both essentially do the same job as long as you’re only powering the board itself. If you want to attach a 12 volt inverter and power the inverter from the board, you must use the battery connector though to prevent 12 volts from potentially getting back into the FTDI circuitry. (that part is 5 volt only)