Proper way to use the reset pin

Hi! Quick question: I connected three arduino's reset pin into one and than to the GND via a button. I understand that to reset the Arduino I need to ground the reset pin through a resistor. With three arduinos, what kind of resistor should I use? Also, is it okay to run an LED light from the reset pin to the GND as a power indicator? I tried it and it seems to work. When the button is pressed, the LED turns off, else it remains on.

I understand that to reset the Arduino I need to ground the reset pin through a resistor.

No you just need to ground it. Pulling it up through a 10K resistor is a good idea.
Just connect all three together an ground through a button.
However that might stop the auto reset working but try it.

Note there is already a pull up resistor on the arduino board.

A small (100R) series resistor with the reset line towards ground is not uncommon, sometimes its done to be 'nice' to a larger capacitor on the reset line so not to discharge it too aggressivly. Other reason is to filter glitches towards the reset line when the wires to the reset switch are long or running past 'noise' circuits.

Regarding the LED on the reset pin, I'm assuming that like you say a led-lamp so it has a build in resistor. If the led-lamp current will also go through the pull-up resistor on the reset line and in effect make the voltage on the reset line lower with the risk of resetting your arduino. Maybe you could ellaborate a bit more on this because it sounds risky.

I would dispute all those reasons on the grounds that they are wrong.

to be 'nice' to a larger capacitor on the reset line

There is no capacitor on the reset line. Do you mean the auto reset?

to filter glitches towards the reset line when the wires to the reset switch are long or running past 'noise' circuits.

Wrong this will do nothing to improve the noise immunity as it only is a factor while the reset button is being pressed it will not affect the normal running of the machine one little bit.

Also, is it okay to run an LED light from the reset pin to the GND as a power indicator?

No problems with that as long as there is a resistor from the reset line to the LED otherwise the on voltage of the LED will drag down the 5V that indicates the running state of the processor.

True , for the arduino board the series resistor is not needed, what I should have added is that it is not uncommon for uController/processor boards/circuits in general. I was trying to find a reason why dominicfhk thought a resistor was needed.