How to properly wire a five color RGB LED strip?
There are no resistors visible, and 2 types of LEDs:

2 pin with one chip
4 pin with 2 chips

LED chips aren't the typical color of white LEDs (ie, some tone of yellow phosphor) so they're not white LEDs. There's some red since you managed to light them, so it's probably RGB. All the LED pinouts support series connection, so you probably got series strings in parallel.
If you don't want to take it apart, I'd recommend using a 12V or more power supply with a 22k resistor in series to limit the current to a value that is safe for LEDs both forward and reverse, then trying all the pin combinations.