Electronic – arduino – Problem with arduino controlled constant current supply with MOSFET/NPN

The two are really the same, functionally. Both work by regulating the voltage over R2 to about 0.6V, what it takes to forward-bias the base-emitter junction of Q1. If the voltage over R2 increases beyond this, Q1 begins to pull down the gate/base of the other transistor. But it can't do this too much, else there is no current in R2, and nothing to forward-bias the base-emitter of Q1. Thus, the circuit achieves equilibrium.

The idea is then that since the LED and R2 are in series, their current is equal. If you can make 60mA in R2.

This is only approximately true, of course, because R2 and the LED aren't exactly in series. In both cases, errors are introduced by the base currents of either transistor. Fortunately, the current gains are very high, so these errors are negligible. I doubt there is any practical difference between the circuits, so selecting based on what you have on hand sounds good to me.

However, if your goal is 350 mA in the LED, then R2 needs to be \$ 0.6V / 350mA = 1.71\Omega \$. You might want to use a 1/2W resistor too, since you are pushing your luck with a 1/4W: \$ 0.6V \cdot 350mA = 0.21W \$. Be sure the transistor you select for Q2/Q3 or M1 can handle the power it must dissipate as well, which will be 12V, minus the 0.6V in R2, minus the forward voltage of your LED, multiplied by 350mA.

If somewhat more than minimum cost is acceptable you could implement a buck converter per LED. These could be very crude and low component count and not a vast price. Each converter would be able to set a current that was proportional to a PWM or analog input. Probably under $1 per channel.

Your 3 x Power supplies plus resistors idea is acceptable BUT can be improved in performance (with increased cost) by making it 3 x power supplies plus constant current sources. The constant current sources are not much different from the buck converters above but with no inductor.

Example. Imagine that your lowest Vf LED (probably red) has a Vf range of 2.0 to 2.5V.
You will need at least 2.5V supply plus headroom. Say 2.7 V.
Minimum efficincy will be when Vf = 2V = 2/2.7 = 74% BUT this is worst case Vf and in many cases the Vf range is lower.
eg say VF = 2.2 - 2.4. Vsupply = 2.6. Zmin = 2.2/2.6 = 85%. Boost supply to 2.8V for more headroom. Zmin = 2.2/2.8 = 78%
Again, this is worst case, so acceptable.

A buck converter will improve on the above.

One solution of zillions. Rsc not needed. Rather than sensing output voltage, LED current is sensed. This C had Vref = 1.25V = far too hight. It can be reduced to 0.6V effective using a diode but still too high. More modern ICs has a lower Vref OR a 1/4 IC section can be used per channel. PWM per channel then becomes 1 x SOIC MC34063 + 1/4 LM324 + a few resistors and a catch diode per channel. May be cheaper than more modern IC.

This circuit, from sound.westhost.com may be suitable.
I independently arrived at a similar version - works reasonably well. This is voltage feedback. Can be made current feedback. Vsense here = Vbe of Q1. This can be reducued with care.