Why do some wire gauge ampacity charts state vastly different current ratings?
Notice that your first link gives two ampacity values for AWG 10 wire. For power transmission applications, the rating is 15 A. For chassis wiring applications, the rating is 55 A. These ratings (and the third one at your other link) are based on different assumptions of convective cooling available and allowances for wire self-heating. 

For example, the first link says


  The Maximum Amps for Power Transmission uses the 700 circular mils per amp rule, which is very very conservative. The Maximum Amps for Chassis Wiring is also a conservative rating, but is meant for wiring in air, and not in a bundle.


You should use a rating consistent with how you're using the wire and the cooling environment around the wire in your application. You should also consider resistive voltage drop along the wire if you find one of the higher ratings appropriate for your thermal environment.