The author of this article seems to have some pretty specific clarification of any grey areas and interpretations, although no documentation of sources other than links to main landing pages.That is why there's an exception for up to 440 pounds of MOT (materials of trade), and an exception for items that are not "in commerce."
https://www.phmsa.dot.gov/sites/phmsa.dot.gov/files/docs/MOTS brochure 2007_10_02.pdf
Removing the Confusion Over Scuba Cylinder Transportation - SDI | TDI | ERDI
I read this as saying that for a shop/instructor, the Materials of Trade exception applies for cylinders that they will personally use up to 440 lbs, but if transporting any air at all for students then there is a "bill of lading" paperwork requirement of sorts for transporting under 1,000 lbs before getting into the need for CDL.