So this average current is the same when calculated as e^-R*t/Ldt ?

The basic operation of the circuit, with the coil meter, is this:

1) when Q1 is turned into a diode (S2/S3 closed and S1/S4 open), the inductor is charged up. During this time, the coil meter is by-passed by S3.

2) when Q1 is turned "off" (chocked off maybe a better word, when S2/S3 open and S1/S4 closed), the inductor continues to flow current, but at an exponentially lower levels (the inductor being discharged). The current will flow through the coil meter. Due to its inertial, you are measuring the average of that current, which is a function of the inductance -> why it can be used as an inductance meter.

Using an inductor in place of the coil meter and measure its current does the same, as the voltage across the inductor follows the same di/dt that's determined by the DUT.

Alternatively, you can use a resistor in place of the coil meter. The voltage across that resistor is proportional to the current, and you can average that, with a diode + r/c network: put a 1k resistor in place of the coil meter (can be lower). From the top of that resistor, run a r//c (10k // 4.7u, or there abouts. The resistor can be even higher but not smaller), from the bottom of the r//c pair, run a diode (1n4148 or the likes) to the bottom of the 1k resistor. So this diode + r//c is paralleling the 1k resistor.

Over a small period of time, the bottom of the r//c pair will accumulate a voltage (differential to the power rail) that is proportional to the inductance of DUT.