I'm not sure about the history of the terminology --
but the "moment of inertia" is what mathematicians would call the "second
moment" of the mass distribution function, viz., INTEGRAL [ X-squared
RHO(X) dX], where RHO(X) is the position-dependent mass density of the
object in question. ("Second moment" because of the second power of X.)
The "zeroth moment" of the distribution, INTEGRAL [ RHO(X) dX] is just the
total mass; the "first moment" of the distribution, INTEGRAL [ X RHO(X)
dX], when divided by the zeroth mass, gives the X-coordinate of the
center-of-mass or center-of-gravity when the object is placed in a uniform
gravitational field. (In the last case, X is the "moment arm" or in
terminoilogy used in discussing gravitational torque, the "lever arm".) One
can, of course, compute 3rd, 6th, 19th, etc., moments of any distribution,
and these higher moments are of little interest to physicists but of some
interest to statisticians.)
The second moment can be computed using X-squared, Y-squared, Z-squared,
XY, YZ, or XZ for a distribution function RHO(X,Y,Z), producing the moment
of inertia TENSOR with six different values filling a symmetric 3-by-3
matrix. (What makes it a tensor is how the values get transformed when we
pick a different set of axes at some arbitrary orientation in 3-D compared
to the original axes.)
The term "moment of inertia" is a blending of the mathematical notion
described above with the physical notion of its significance, namely, a
measure of the resistance of the extended body to angular accelerations
about specifiied axes.
"Rotational inertia" is a more intuituively physics-oriented description og
the same thing. I would not stress too much about this terminology -- they
mean exactly the same thing, and in introductory physics courses
(especially non-calculus based) the term "rotational inertia" is perhaps
more intuituve and less confusing.
Peter Vajk
St. Joseph Notre Dame High School
Alameda, CA