In the dark, with my tri-focals misaligned, I suspect what I was seeing was the relay and the socket, now that I think about it.
Kind of reminds me when I was a teenager, with no experience with women, I saw a young woman in sheer summer weight wear, and wondered why she was wearing two bras, not realizing one set of straps was for the slip - normally covered, but exposed on her otherwise bare shoulders.
All three relays are exactly the same, same part number on each one. You could try switching them around and see what happens. I have had a very similar problem, the dreaded click I fought for two years on and off. Remove and scotch brite all connections on the battery and the starter - and+ that will work for a while maybe.
Final solution for me was a new starter solenoid. Starter spins three times faster then it ever did, even when it was new.
Took a solenoid off of a new Toyota starter cut down the plunger to size and bolted it up to the triumph starter and bam, no problem.
If you need a part number let me know. Also there is a guy on eBay solenoid rebuild kits. Good luck.
Correct, the two on the right side are both different and also different than the ones on the left. I need to look them up, not sure what they switch. Their is another different relay under the tank for the aux lights.