Is it possible to measure lightning discharges as Nikola Tesla?
sensitive receiving instruments arranged so as to be capable of responding to electrical disturbances created by the discharges


A lightning strike is an enormous current flowing for very shortly. That (thanks to Ampere and Maxwell) we know to induce an electromagnetic wave with a very high bandwidth and very high power.

So, any radio receiver these days is more than capable of "detecting" such discharges (simply by malfunctioning for a short time).

What Tesla had was probably even simpler: Close enough to the discharge, the change in E-Field might be drastic enough to simply make a spark gap across a simple coil spark.

Really, what about googling "lighning detection"? There's large-scale citizen science projects where people contribute their own lightning counts to a large central database.