What is the minimum electric field strength,that the induced electrical tester led begins to turn-on?
I would suggest that the device you show is a very poor electric field tester.

The AC electric field tester (a quite high sensitivity device) has actually not been around all that long and there are a couple of detailed patents for them. This one shows a generalized schematic so it might help you.

You can see that it's based on a CMOS inverter (something like a CD4069), so the input voltage sensitivity is about 1 V. The sensitivity in V/meter is actually orientation as well as field sensitive. It would be tough to work out without access to the PCB layout, but you could consider it to be roughly defined by the sum of the PCB track length either side of R1. If we guess at say 1" track points ....then at 1 V sensitivity that would be approximately 39 V/m.