Electronic – Light up an LED with an NPN transistor in active low

That quote is just wrong.

That circuit is correct, but not for your application as it will require about 8V minimum input to give 6V at the output.

It works by supplementing the output current of the 78xx (7806 for 6V) regulator if the regulator takes significant current (about 600mA) with additional current through the PNP transistor.

Your Lipo batteries will not have a constant output voltage, they will probably vary between 8.4V when fully charged down to about 6V or even less when discharged.

What do you need the 6V for? How much current do you need? What is the acceptable voltage range for the load?

It's not clear why you have chosen a PNP transistor to begin with - I assume it's because you had to start with something. Throw it in the bin; get some MOSFETs, Darlington transistors or beefy NPN transistors (in order of current-conducting capacity, highest to lowest, generally speaking).

The following diagram uses IRF510 MOSFETs, which are standard off-the-shelf types and are sold on all manner of hobby electronics sites in one form or another - maybe IRF540 or 530 instead. It can handle currents of 4A or more (read the datasheet!) and requires very little in the way of external components, as you can see:

You don't need the 3.3V rail; it's of no use. If you can provide some specs for the 12V LED cluster (we accept datasheets or cold, hard numbers) then somebody can suggest a suitable device to drive it if you have trouble sourcing certain components.

EDIT:

In the Falstad circuit you had the forward voltage of the LEDs too low (11V). That simulator is very basic at best, and so was only dropping 11V across the diode leaving 1V at the gate of the second MOSFET. The emmended can be viewed here.