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It seems to me that the "dimension" of a measurable quantity answers the
question "What are we measuring?"; a question of KIND or CONCEPT - often
based on some conceptual model of the phenomenon under consideration and
its interaction with the measuring instrument.
...
I would classify length, mass, time, angular opening, charge, etc as
"dimensions"; and meters, kilograms, seconds, radians, and coulombs as
"units". The radian is not dimensionless anymore than the centimeter is.
They are each the unit of a dimension (angular opening and linear
extent). The fact that the size of an angular opening can be
measured/defined (in radians) by dividing two lengths is neither
bothersome nor unique.