It seems that they are just LEDs. LEDs are seriously bright nowadays. They are normally rated for a certain maximum current. LEDs should be supplied from a current limited supply, not a fixed voltage.
30 seconds of a significant overload won't always blow an LED. Close up, white LEDs can be seriously bright and the human eye isn't good at comparing brightnesses for small, bright dots.
I assume that strip is 8 LEDs, as you mentioned 8, although only 6 1/2 made to the picture. You mentioned the manufacturer's website but didn't put a link to it.
A strip of LEDs may have the individual LEDs in series or parallel, or a mix of series and parallel.
In series, the current is the same in each LED, but the voltage is the total of all of the voltages, so if the LEDs are in series, I would expect at least 3 V per LED, and therefore 24 V, and it wouldn't be bright on a 20 V supply. However you don't say what the 20 V supply is, and many will be more like 24 V with little load.
If the LEDs are in parallel, 8 lots of 350 mA is 2.8 A, and your power supply might not be able to provide that. A 20 V power supply that can supply 2.8 A is rated at 56 W, so it would be similar to a laptop power supply. If you had a power supply that was limited at 0.5 - 2 A, that would give 60 mA - 250 mA per LED, which would be really bright if looking directly at them.
Do you have a multimeter to measure voltage / current? Please tell us where we can see the datasheet for the LEDs or the strip of LEDs, and what power supply you are using.