It might have been out of fashion after 1914 for a clean cut cowboy but when you look at true 19th century photos of working cowboys, wearing pants inside the boots was the norm and it actually still is today if you go to a farm or ranch where people are actually working.

It only makes sense just as we wear our pants inside our rubber muck boots or even our leather work boots today when working in the mud or around livestock. Mud and manure wash off of boots easier than out of pants and cowboys of that time were wearing those same pants for weeks or months at a time. Plus thorns and sticker bushes don't snag leather boots like they do pants.

Hmmm, my born in 1914 Montana Cowboy/Farmer father claimed only Roy Rogers and Texans stuffed their pants in their boots.  On the outside keep stuff from getting into the boots.  If mucking the muddy shitty stable pull pants up and stick on the notched top so they don't fall down and get dirty