Electrical – MOSFETs acting strange on different power supply

You need a commmon ground between the two. Probably your bench supply (-) output is grounded and the other ones are not, and you are getting a ground through the USB cable and your PC, so the grounds are connected through the 3-pin wall plug when you use the bench supply. If you unplug the PC it will probably stop working. You may wish to use a 1K resistor between the two grounds, just in case.

Also (and this is not the cause of your proximate problem), your BUZ11 is really an inappropriate part for that service. It's not a logic-level MOSFET and is only guaranteed to fully turn on with 10V Vgs. You should use a MOSFET that has an acceptable specified Rds(on) with 3.3V or lower Vgs. The particular sample of BUZ11 you happen to have may sort of turn on enough typically, but that is not good engineering at all.

They probably will, in my opinion, but there is no guaranteed with such a MOSFET. The key number is the guarantee of Rds(on) at the lowest gate voltage shown in the "Specifications" section. Vgs(th) is the guarantee that the MOSFET will be almost entirely off (250uA). There is a region between Vgs(th) and your operating point and you want the voltage across the MOSFET to be small or it may get too hot and your LEDs may not be at full brightness.

In this case, the Rds(on) is guaranteed to be less than 0.039 ohms at 25 degees C Tj with 4.0V Vgs. The transfer characteristics curve is only a typical characteristic and is not guaranteed so it is bad engineering to use it for your particular purpose. It may be appropriate for a hobby project to give it a try, of course.