My DIY RGB LED panel powered by 9V battery works but certain colors like blue, white, and purple don't work
It is a bit hard to see from the picture, but since each LED has six pins, and knowing that it is meant for 12V supply, it is extremely likely that the LEDs might be arranged so that there are three parallel chains of three LEDs in series.
The blue LED requires approximately slightly more than 3V voltage drop per LED, and since there is 3 in series, the three blue LEDs require slightly more than about 9V to work. A 12V supply with small series resistance and transistor to drive them would work.
So basically, the LED board is meant to operate with 12V supply, and it can't work with a 9V supply.