Two different current ratings for power cables?

The ampacity of cable depends on ambient temperature, insulation limitations, and installation conditions. The "power transmission" rating in your powerstream link is not a regulation, it is a conservative estimate based on 700 circular mils per amp. For installations covered by the NEC, use the NEC rating if installation conditions are met, derating if necessary for other conditions. The Qualtech cable you referenced is a cord set with a manufacturer's rating. The manufacturer's rating cannot be exceeded.

In my opinion, if we could speak about "rating" then only the conductor maximum permissible temperature for continue loading-no duration limit-or for short time or for limited time, may be considered rated.
For xlpe insulation, for instance, continue maxim temperature is 90oC, for short time 250 oC and for limited time as "Emergency Overload" 130oC. So, this will be the temperature of the conductor, in contact with this insulation.
The temperature depends on the losses produces in the conductor-proportionally with conductor resistance and the current and the environmental medium temperature. When the conductor losses are evacuated towards the environmental through different layers of insulating material -conductor insulation, jacket, serving, ducts, concrete or air, earth and other, the temperature drops with the multiplication between losses and the thermal resistance of the layer and eventually gets the environmental temperature-air or earth for instance.
Then, the ampacity of a cable is different for exposed in free air, in conduits, on cable trays, outdoor or indoor, underground burry directly, in conduits or ducts, in concreted duct bank and other.
Even, the conductor resistance it is not a constant and depends on material type-copper or aluminum or else, temperature, frequency, number of parallel cable and other.

You referenced two completely different application listings for wire Amperage and got different numbers. The Qualtech 313010-01 is a power cord, and 16 AWG for use over 6 feet is perfectly fine and typical for a power cord. As jghrist pointed out, the amperage of wire depends upon ambient temperature, insulation limitations, and installation conditions. I would add to that LENGTH (don't want too much voltage drop).

The power transmission reference you used is a guideline for LONG LENGTH applications. A 16 AWG wire has 4.09 Ohms/1000 meters at 25C (increases with temperature). So if you tried to pass 16 Amps through a 1000 meters of 16 AWG wire you would have 65 Volts of line drop. An unacceptable current for 16 AWG power transmission over long distances. But if you passed 2.3 to 3.6 Amps (by the chart) you would have 9.4 to 14.7 Volts of line drop.

Typical example from my area of experience. I design power inverters. I might pass 50 to 70 amps through the lead of a TO-247 MOSFET that is 1/2" long. The MOSFET lead has the copper area equivalent of about 16 AWG wire. But this current is fine for this application.