Determining how much current I can actually run through my Cat6 24AWG Cables in my confirguration?
The Table you show is generic, not taking into account wire length or insulation material.  


  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.


The maximum current you can put through a 24AWG conductor depends on the insulation material of the cable.

A lot of CAT cables use HDPE Insulation to insulate conductors.  Jackets vary from PVC to LSZH to protect cables.  



Ignoring the jackets.

HDPE is high-density polyethylene.  From Current Carrying Capacity of Copper Conductors, has a maximum temperature of 90°C.  At an ambient temperature of 30°C, 24AWG has a maximum current of 7A.  

So 2A for a 10ft cable will not be a problem for 24AWG wire.