You are correct, the distinction here is between unitless constants and constants expressible only in terms of arbitrary units.
ex:
The fine structure constant 7.29735257×10^−3 is independent of any measurement system and takes the same value when computed in any of them. The speed of light, however, cannot be expressed without prior definition of units of time and distance. 299792458 m/s is equivalent to 1.8026175×10^12 furlongs per fortnight.
ex: The fine structure constant 7.29735257×10^−3 is independent of any measurement system and takes the same value when computed in any of them. The speed of light, however, cannot be expressed without prior definition of units of time and distance. 299792458 m/s is equivalent to 1.8026175×10^12 furlongs per fortnight.