The Texas Rangers were a limited force following the Mexican War, flowering briefly in 1858-1859, fighting Cortina's band of raiders alongside the U. S. Army. By the time of the Civil War, the force was small. Members individually rushed to Confederate colors. The Eighth Texas Cavalry, known as "Terry's Texas Rangers," was a mixed bag; the founder, Terry, never had belonged to the Texas Rangers. Your mandatory Texas history probably had as much myth as my mandatory New Mexico history.

There is a lot of fakelore about the Ranger belt. Not until the 1930s was dress provided. Until then, Rangers bought their own clothing (Handbook of Texas Online) and they bought to suit the job. Somewhere on the internet is a website by the Ranger Society showing one of the several types of fake buckles. A fancy silver one with the star, is sometimes worn by retirees. It is not wide See Buckle

The "Ranger belt" was a standard type of the times, width variable but generally 1 1/2 inches wide, with a small subsidiary 'belt' (the billets),' which carries the buckle, stitched onto the main belt. The buckle generally was no wider or only slightly wider than the base belt; its horseshoe or "U" shape was originally English (or European). Perhaps some Rangers wore them. See Ranger Belt for usual proportions, or, a better picture Belts and scroll down to the ranger belts, showing buckle and keepers.
A buckle any wider is worn only by foreigners (non-Texans), line dancers or rhinestone cowboys (and rodeo or show types).