I have a bunch of spare wires with female and male pin ends for my bike, I used those with my alligator clips to do my test.Ok, that is great info! While the battery disconnected and battery connected results appear to confirm all is good, I am very troubled by the fact that you could not hear the headlight relay click when it was energized. (terminals 15 and 9 energize relay "A" which is your headlight relay.
What I would like you to try (if you can) is to perform the same test of applying battery voltage to 15 & 9 while a load is applied to 5 & 10. Ideally, use your headlight bulb as the load. To do this however, you will need a power source for the switched side of the relay (5 & 10). This will of course be a bench test with the advantage that you have removed all other wiring, switches and connectors that are on your bike. When you do this test listen carefully for arcing, sparking or buzzing from the relay contacts and of course observe if the headlight bulb come on nice and bright and steady.
I am suspecting that your relay contacts have fused together to the extent that they won't conduct until the magnet draws them in tighter. Just a theory of course but it would explain why no click yet relay appears to work.
I am not sure how you did the first batch of tests, but my preference is to use jumpers that have proper connectors instead of alligator clips that can accidentally short out. So if you have, or can get the appropriate male or female connectors, this makes your testing much safer and more secure.
So for the next test I'm kinda confused. Do you want me to rigged up a bulb 5 and 10? While I power up 15 and 9?