First, the link:
Very clever on Theodore Roosevelt's part to have ordered this, as it is in .38 Long Colt, the same round for which the Colt Model 1892 was chambered. Thus, it is obvious that Roosevelt did take delivery of this just prior to training Rough Riders in San Antonio, Texas. Surely, he would not have ordered this in .38 Long Colt otherwise, due to the poor ballistics of the .38 Long Colt round: https://en.wikipedia.org/wiki/.38_Long_Colt
The cartridge's relatively poor ballistics were highlighted during the Philippine–American War of 1899–1902, when reports from U.S. Army officers were received regarding the .38 bullet's inability to stop charges of frenzied Moro juramentados in the Moro Rebellion, even at extremely close ranges. A typical instance occurred in 1905 and was later recounted by Col. Louis A. LaGarde:
Col. LaGarde noted Caspi's wounds were fairly well-placed: three bullets entered the chest, perforating the lungs. One passed through the body, one lodged near the back and the other lodged in subcutaneous tissue. The fourth round went through the right hand and exited through the forearm.
Antonio Caspi, a prisoner on the island of Samar, P.I. attempted escape on Oct. 26, 1905. He was shot four times at close range in a hand-to-hand encounter by a .38 Colt's revolver loaded with U.S. Army regulation ammunition. He was finally stunned by a blow on the forehead from the butt end of a Springfield carbine
As an emergency response to the round's unexpectedly dismal performance, the U.S. Army authorized officers to carry M1873 Colt Single Action Army revolvers, chambered in .45 Colt, and issued from reserve stocks."
All other Rough Riders were issued Colt Model 1892 revolvers: https://en.wikipedia.org/wiki/Colt_M1892
Ordering an engraved revolver was a bit of showmanship on Roosevelt's part. However, surely he could have ordered an engraved Model 1892 revolver, if desired, or purchased one, surely. I suspect the reason for this is because Roosevelt knew that the Smith & Wesson New Model No 3 revolver was the superior of the two revolvers in loading and unloading on horseback. Simply break open the revolver and load six rounds or eject six spent cartridges. The Colt Model 1892 was far superior to the Colt 1873 in loading and unloading rounds, as the former required pulling the hammer back to half cock and rotating the cylinder six times to load or eject spent cartridges. At least the Colt Model 1892 contained a cylinder that opens and closes. On horseback, I imagine this would bounce about a bit, which is disadvantageous relative to the more stable cylinder of the Smith & Wesson New Model No 3.