A person could simply ask himself why did the revolver companies include a spare cylinder with their cased guns, and/or make spare cylinders available as a special order?
Folks wouldn't buy a spare if they didn't think that they had a use for it.
There's such a thing as a retreat, where a cylinder swap is still faster than reloading fresh rounds, especially if paper cartridges aren't available.
Companies did include spare cylinders with very few guns, cased ones particularly, and on demand, but there's no evidence whatsoever
I could find that it was a widespread practice, least of all that it was standard issue for cavalry troopers.
Why would one want a spare cylinder, if not for having a preloaded cylinder to swap?
Two very simple reasons (much simpler than something like swapping preloaded cylinders with all the issues doing so implies) that have been stated for having a spare cylinder, in situations when you had time to do things like maintenance, but you weren't completely safe and therefore wanted your revolver back into action asap, which are:
1) Having a spare available if the main one had any problem (like a broken nipple or something that rendered it inoperable, like the gun being dunked in water that wetted the powder): instead of having to wait a long time for another cylinder (in case of a mechanical problem), or having an inoperable gun until you fixed the issue, like having to draw all the charges, you'd have a spare one to use while you tried to fix the inoperable one.
2) Having a cylinder ready while cleaning your gun. The cylinder is the most time consuming part to routinely clean (short of disassemblying the gun completely): one could clean the barrel and frame in very little time and reassemble the gun with the spare cylinder, having a useable firearm while he took the time to clean the other cylinder.
This said, yes, Pony Express is the single most plausible case where it made sense to have a
loaded spare cylinder instead of a second gun. Everything in the pony express equipment was kept as light as possible, down to the satchel buckles... even the rider (which explains Billy Tate).
I think that damoc provided a good lead which now the ball is in your court to try to do some of your own research.
It's okay to ask, but since you're the one that wants the info, what have you done to try to follow up on the lead?
Maybe you can search for a book about the incident, the person or about the Pony Express.
We're here to help you, and also to help guide you to help yourself.
The internet provides many avenues for research, but it also has its limits as far as providing direct proof.
If there were such a book where it was mentioned about the Pony Express riders or other XYZ persons carrying a loaded spare cylinder, and/or circumstantial or other evidence that it was used in a gunfight, then there's bound to be other issues such as the validity of references or footnotes, and where the author found the information.
There's always going to be folks who doubt whether something happened or not.
The veracity of some events are simply based on the preponderance of the evidence.
There's a misunderstanding, I'm not asking you to do my research for me: I was just asking is if someone who already did it had any useful information to share. I did my homework, but unfortunately I couldn't find any primary sources (but I found a lot of other people that published articles or posts indicating the lack thereof). So, this being a forum where there's a lot of knowledgeable people, I asked if someone had any info.
For example, I found
"The Pony Express: Bringing Mail to the American West (Milestones in American History)", which again is not a primary source, and it's also quite expensive (even in Kindle format), while containing a single reference that may or may not prove useful. If someone has it, Could he share what it says at page 138 on the topic of Billy Tate? 
that's why I excluded the Pattersons from my question: with those it was most probably done.
