I'm still skeptical of whether any of the 20th Texas Cavalry (dismounted) were sent from Boggy Depot to the site of the battle in Pontotoc County. What is the source of "the history" of the 20th that says they were there?

The scenario you describe of some number being given mounts and sent to the vicinity is reasonable but I have found no credible source to suggest that occurred. All the claims I've seen that they were there come from the incorrect assumption that the Feb. 13th battle occurred at the Texas Road crossing of Middle Boggy near present Atoka. I've been told (can't find the source of this now) that some/all of the 20th TX were sent to the Texas Road crossing along with some of Quantrill's men in case an advance was made from that direction. Combining this with Phillips' statement that the opposing force at the Feb. 13th battle was "Seminole, Choctaw, and Texans" leads to the assumption that the "Texans" were the 20th TX Cav. There are no Confederate accounts, that I have found, that suggest that the 20th Texas was there or moved there. The Confederate accounts only mention Jumper's Seminole Battalion and Nail's Company, which is apparently the company of Capt Jonathan Nail of the Chickasaw Battalion; the Chickasaw Battalion being composed of Choctaw, Chickasaw, and Texans organized for service in the Chickasaw Nation.

Without first hand accounts or some contemporaneous documentation that states the 20th Texas Cav were mounted and sent to join with Jumper's Seminole Battalion and/or Reynold's Chickasaw Battalion, I would argue that they remained dismounted at Boggy Depot.