I am seeking the best method for testing to see if a diode type cut out is good or bad while mounted on the generator.
If it's a diode a simple ohmmeter should do the trick, wouldn't it? Lots of resistance one way, almost none the other way.
Tim has it right. To check it out totally I would remove it from the generator. You then want to measure the resistance from the forked generator terminal to the screw terminal in one direction using an ohm meter and then reverse the probes and measure the same 2 connections in the other direction. It should be a very low resistance in one direction and a very high resistance (near infinity) in the other direction. You then also want to make a resistance reading between the screw terminal and the base of the cutout in both directions too to make sure there is no leakage discharge path for the battery power. This should be done on a high range of your ohm meter. Typically such a leakage path if severe will show up on your ammeter when the power is connected but even a high value of resistance is suspicious and should be checked out. Usually a failed diode will fail shorted so that it measures a very low value of resistance in both directions. Leakage to the base sometimes takes place because of a lot of heat inside the cutout has caused insulation breakdown and when replacing the diode only you might not think to check that.
It depends. Some multimeters (analog or digital) do not have enough voltage to accurately test diodes. That's why most digital multimeters have a special diode check function.
All that being said if your meter shows high resistance in one direction and low or near zero in the other, you should be OK.
In other words, without a diode check function you could test a diode as bad when it isn't, but if it tests good, you should be OK.
To avoid the digital nightmares I bought an old Simpson 260 analog on eBay which is what I grew up with in the military and became fairly proficient with it. It stays in my T-Barn where I do my major troubleshooting.
I still have a couple of digital paperweights I carry around because MOST of the time they fulfill the need.
I agree with you and also bought a 260 years ago as I used one a lot in the military. I do however like the digital when doing voltage drop test. 99% of the time however, I use the Simpson. As a caution, remove your batteries when not in use. I'm guessing most ruined meters are from battery leakage. Either that or treat it twice a year when you do your smoke detectors with new batteries.