AWG Conflicting Sources
This is not a matter of "trust-worthy", it is a matter of understanding of basics of electricity. One should always remember that any wire is a resistor. As a resistor in a circuit with current flowing, it 

(a) dissipates energy, and therefore produces Joule heat, and 

(b) voltage drops along the wire. 

These are two major restrictions imposed by wires in DC applications. That's it.

The wire gauge system is based on restriction (a), to what extent the design can tolerate increase of wire temperature. The temperature raise depends, in turn, on ambient conditions, whether it is assumed to hang in open air, or moving air, or encapsulated into equipment or thick insulation jacket. That's why different sources might differ in assumptions about the acceptable level of temperature increase, or duration of current pulse, etc., and give slightly different results.

The second effect is (b), voltage drop, so in voltage-sensitive application (like LED steep forward voltage), the drop might have a strong effect. According to Wikipedia, 20 AWG wire has resistance of 10 mOhms/foot, so 15 feet will have 0.15 Ohm resistance. With 2 A current it will result in voltage drop of 0.3 V one way, or 0.6V less at the load (accounting for ground return wire drop). For a 12-V setup this sounds like tolerable, unless there is a design error in LED hook-up.

The same source indicates that a 20 AWG wire can sustain 7.5A with 20 C raise in temperature, so 2 A shouldn't be any problem either.