I read "Feeding power to Arduino: the ultimate guide" and I have this 5V "LED power supply". The guide says you can plug it in to the 5V pin if "having an external stabilized 5 V source".
Does this device count as stable? Also, why does Arduino mainly accept a 12V input - I'm guessing this is so that it can stabilize it?
I won't link the eBay product page, as few details, but title is "5V Lighting Transformer Switching Power Supply AC 110V-220V Cooling LED "
Voltage output measured is 5.65
Need more information.
When you put power into the 5v pin, do not power the Arduino via the USB connector.
5.65v is too high.
5.65V applied to the 5V pin may damage the on board regulator. You could use that supply if you put a diode (e.g. 1N400x) in series with the positive lead, which will drop about 0.6 - 0.7 V.
Also, why does Arduino mainly accept a 12V input
The Arduino has a built in 5V regulator, and works best with 7 to 9V applied to RAW or the barrel jack. 12V can be too high under some circumstances (for example, when several pins are connected to LEDs).
Yes, that is a perfectly appropriate device to power an Arduino via the 5 V pin. That is in fact, the proper way to power it.
The regulator on the Arduino UNO/ Nano/ Pro Mini/ Mega2560/ Leonardo/ Pro Micro has very little heatsink, so will not pass very much current (depending on the input voltage and thus, how much voltage it has to drop) before it overheats and (hopefully reversibly) shuts down. It is essentially a novelty provided in the very beginning of the Arduino project when "9V" power packs were common and this was a practical way to power a lone Arduino board for initial demonstration purposes. And even then it was limited because an unloaded 9 V transformer-rectifier-capacitor supply would generally provide over 12 V which the regulator could barely handle.
Nowadays, 5 V regulated switchmode packs are arguably the most readily available in the form of "Phone chargers" and switchmode "buck" regulators such as you have are cheap on eBay so these can be fed into the USB connector or 5 V pin to provide adequate power for most applications. Unfortunately, many tutorials or "instructables" are seriously outdated or misleading and have not been updated to reflect the contemporary situation.
As you already have that fully-regulated power supply to hand, and particularly if you propose to add any other devices that will require 5 V, to the basic Arduino, then it will be a most appropriate power supply. Note it has an adjustment terminal to get the voltage to precisely 5 V, not that it is all that critical as 5.65 V is well within the working voltage of the ATmega chip. 
