The '7135 is a linear current regulator, all self-contained in one tidy little package. Each one contributes 350 mA to the output. Want 700 mA? Use two. Want 2.1A? Use 6.
There's typically a capacitor or two on the board to support the '7135s. If there are any other electronics, it's a microprocessor to provide modes, and some support stuff for the micro.
Those boards in your first link are set up to accept up to four, with only two installed. It has a single mode - full power all the time. The other one has eight installed, and a micro for modes.
If that's a buck-type switching regulator you have (which is consistent with the coil being there), and only 71% efficient, it's a poorly designed one, or perhaps a failed one.
Most meters set to measure DC will measure average. Some have peak-capture feature, but they have limitations on how fast a signal they can capture. Few if any could capture the peak of a regulator's switching signal. Set to AC they will try to measure RMS values, but again the signal is too high frequency for most to measure.
There is one thing that may partly explain how you got a lower than expected efficiency. When you insert the meter in the input line to measure the current, you introduced additional resistance in the circuit. This would cause a voltage drop at the input of the driver, which (because it is a switcher) causes the input current to rise. Unless your meter has unusually high resistance, this likely doesn't account for more than a few percent, given the numbers you posted.