"The distinction should be "open" and "commercial" (and maybe not even that!)"
No the distinction is between open source and closed source. The other orthogonal distinction is commercial versus non-commercial. You can have the four combinations:
open source, commercial: RHEL etc.
closed source, commercial: MS etc.
open source, non-commercial: majority of sourceforge ?
close source, non-commercial: card ware, freeware, xxxware, etc.