Electronic – Help to build a circuit to trigger event only after input signal changes from “1” to “0”

You definitely don't want an instrumentation amp. Have you paid close attention to your timebase? You are looking at pulse widths on the order of 20 nsec, and you just won't find an IA that will handle that. Since you seem more comfortable with positive thresholds, take your pickup wire and swap the ground and signal. (You can do this since the wire is galvanically isolated from the original source, and your choice of ground and polarity is entirely up to you.) This will give you a larger positive pulse. I'd recommend a fast comparator with a trigger level in the range of 50 to 100 mV. As a starting point, see http://cds.linear.com/docs/en/application-note/an13f.pdf for a fairly comprehensive look at what you need to worry about. Another possibility is the AD8561.

With pulses this fast, I don't see any reason to go for a Schmitt trigger configuration.

ETA: Although, Figure 42 from the app note does exactly that. And I would never argue against Jim Williams.

You could use a MOSFET driver such as the TC4427 with a 47Ω series resistor. The resistor will sufficiently limit the power to protect the output.

Note that with a 50Ω output resistor the voltage on a high-impedance input will be double that on a 50Ω input.

What you really need depends on the device that takes the signal as input.