Lobo's info is correct as far as it goes. And limited to veg tan leathers such as the byproducts of shoe leathers that we use to make holsters and saddlery (we've long needed a similar word, so we say 'and accessories' but they're simply other forms of carriers). Chrome leathers don't react in the same way at all (look at an old shoulder holster harness). Heisers and Lawrence holsters were routinely oiled for increased fibre strength and for weather resistance.
Bare veg leather, left in the sun, turns red from the ultraviolet. A sun tan or sun burn in every sense of those words. But if the leather has been oiled, the leather instead will darken, and without reddening. Perhaps like a form of sunblock? I don't know the science. I've intentionally done one or the other to create these effects.
I expect that the finish you're admiring, is the latter with a hand- rubbed finish on it and without some sort of sprayed-on finish including but not limited to the leather lacquers. For John's western museum we'd distress the holsters first, as one would with distressed furniture.
Is there a maker you admire, who does western gear (I don't presently)? Then ask him if he knows how to do this for you. Better yet, though, buy the vintage/antique genre. There's a gorgeous double set on auction now, a Heiser.
Which reminds me, that looking through historical references about Sam Myers, that the word 'rig' has a basis other than as a synonym for 'set' (Michael Wayne insisted that we call his Dad's commemorative holster and belt a 'set'): 'rig' refers to how the saddle originally, or holster later, was 'rigged'. That is, how and with what it was set up; with which features and straps and accoutrements etc. So, an Ojala holster would be 'rigged' in one way, and a Keith style to be rigged in another, very different way. Ditto the aforementioned shoulder holster. So, 'rig'.