It is entirely possible to detect lighting strikes with modern (and not so modern) equipment.
Any AM receiver will "detect" lightning strikes. There are also electrometers which can detect strikes as well as the conditions before the strike actually occurs.
Standing waves caused by resonance should also be detectable.
A lightning strike causes a very broadband burst of radio waves. It should cause radio waves in (nearly) equal intensity up to very high frequencies (GHz and beyond.)
It should be possible to use a spectrum analyser on the RF received from a lightning strike. Resonances would show up as peaks at particular frequencies, absorption would show up as dips.
The frequencies are very low, though. Perhaps lower than most RF equipment is designed for.
The patent you link to refers to resonances between wavelengths of 25 to 70 kilometers. That means frequencies between 4.3 kHz (70 km) to 12 kHz (25 km.)
Those are more typically associated with audio frequencies.
This makes an intersting experiment possible:
Use a long wire and a ground wire to receive the signals. A small amplifier may be needed, may be not.
Connect the long wire and the ground to the microphone input on the sound card of your PC
Record typical noise when there is no known thunderstorm in your area.
Record typical noise when there is a thunderstorm in progress.
Compare the spectrums of the two different recordings, and look for peaks or dips during the thunderstorm.
Lightning strikes in the recordings should be visible as "spikes" in the waveform view, and as broadband increases in the spectrum view.
If the peaks are too weak to clearly see, then you may need an amplifier and maybe impedance matching to get more of a signal from your antenna. The amplifier should be fairly undemanding - a JFET preamplifier as commonly seen for microphones should work well.
It should be fairly easy to build and use such a setup. Just a couple of wires, a PC, and some common software. Audacity would work, or Baudline. I'm sure there are many others.
I'd personally use Baudline, and the cross correlation function to compare the recordings.
You could also write your own software, but it is probably better to see what you can do with the available software before you write your own.
Given that Tesla could find the resonances with the (relatively) crude measuring equipment available in his time, they ought to be fairly easy to detect with modern analysis methods.
The above ignores the safety aspects of connecting an antenna and ground to your PC. That's a seperate problem. A long wire pointed upwards during a thunderstorm is basically an invitation for lightning to strike.
If your house has lightning protection, and your antenna is fairly small, then it should be safe enough - the rabbit ears antenna on a portable radio doesn't attract lightning.
A really long wire that stretched from your house a couple of hundred feet up will attract lightning.
Radio amateur operators have equipment for dealing with that kind of problem, so there are solutions available.
For beginning experiments, though, a small antenna should be safe enough and adequate to see if you want to continue.
Note that you are not building an AM receiver. You are just picking up and digitizing the "RF" as it is. No diodes are needed.