I don't know any of the history, but the phrase "moment of inertia"
pertains to the physical (and mathematical) quantity itself. It would
be absolutely clear if it were called "first moment of mass" and
referred properly to a point in space in the same breath. Convention,
however, has blessed "moment of inertia", even without a reference
point (or axis, if one is referring to a scalar quantity), a practice
I actively discourage in my teaching. The practice of Halliday &
Resnick et seq. is a minority practice, as you have found already.
The term "rotational inertia" should refer to the property exhibited
by an extended body to remain in a state of constant angular momentum
unless acted upon by a net external torque*, just as "inertia" is the
property of a particle to persist in a state of constant linear
momentum unless acted upon by a net external force. "Mass" is the
physical quantity which is a measure of the degree of inertia.
Leigh
*Angular momentum and torque should also be referred to a common
point in space, of course