First of all, thank you for the detailed response and giving the source. I love discussing/debating the finer points of drill like this to try and arrive at a better understanding of Civil War drill. So I have some thoughts about this. In school of the company, Casey gives a different explanation of "Rest"
39. If, on the contrary, the instructor should wish to rest the men without constraining them to preserve the alignment, he will command:
REST.
40. At which command, the men will not be required to preserve immobility, or to remain in their places.
41. The instructor may, also, when he shall judge proper, cause arms to be stacked, which will be executed as prescribed in the
S. S.
This version would seem to support that the men leave the formation as they don't need to stay in place. I would say that this version would supersede the school of the soldier and the other was probably just a training tool of some kind like a lot of the things in SoS.