Perhaps a slight change in terminology might help. It is true that
additional terms must be added to F = ma in a non-inertial frame. The
terms can be included on either the left or right side, and many people
may find it easier to think of them as forces, instead of modifying the
acceleration measured in their "rest frame".
To me, however, "kinematical forces" sounds to much like gravitational
forces, elastic forces etc. Perhaps "apparent" or "illusory" forces or
something like that could satisfy both camps.
Al Clark, again.