What's the name for the style of shirt that's associated with Western wear, has two breast pockets with flaps, detailed stitching, and generally pearl or imitation-pearl buttons or snaps?

western shirts

Idk about the whole denim/flannel thing Mighty is talking about but if you search for western dress shirt it will give you places like country outfitters sheplers stetson.. and many more.

Google “pearl snap shirts”. Generally they’re just called pearl snaps.

I just searched for 'cowboy shirt cotton' and got lots of images that seem to match what you're looking for. Lots of variety in the amount of stitching and other design elements too. Hope this helps!

You're probably gonna find them through the material. They're mostly denim or flannel. I just entered "denim shirt men" into my browser and got the shirts immediately. You're going to look at Levi's and Wrangler and the like. But why not shop for them second hand, too. They look much better imho if they're a bit worn. Or please go take a look at your country's fair fashion stores over those big brands.

It can be a bit expensive, but you can get custom ones from Proper Cloth in any fabric they sell. They have a few pre-made designs for western shirts, or you can start with the fabric you like and style it with the western details. https://propercloth.com/shop/search/western?sort=recommended

This one’s called a gonzo western shirt, made in lyocell: https://www.nudiejeans.com/en-GB/product/gonzo-western-shirt-sunburns-blackbrown