IMHO
1. There are many "Work-Energy Theorems" derived from Newton's second
law. Generally, the form to be used is specific to the purpose or the
problem (eg to introduce the concept of energy, or to treat a many body
problem).
2. The term "Internal Energy" has a different meaning in the context of
the treatment of a collection of hard sphere particles than it does in
the Thermodynamic context. The distinction in mechanics is between
energy of the system associated with the center of mass, and energy
associated with motions about the center of mass. In mechanics, rotation
is "Internal Energy".
3. One should keep in mind the most elementary introduction of the topic
of work and energy. For example, starting with a single particle in one
dimension and using (V2)^2 = (V1)^2 + 2aS ,where S is the magnitude
of the displacement from point 1 to point 2, and is small enough so that
a can be considered constant.
Newton's 2nd Law multiplied by S : FS = maS = (1/2)m(V2)^2 -(1/2)m(V1)^2
At this stage FS is identified as the "Work done by F during the displacement
S " and (1/2)m(V)^2 as the "kinetic energy at that time"
This becomes the simplest form of W-E "theorem".
Adding a second force for which FS is a function only of the coordinates
of the endpoints allows introduction of potential energy functions, U and
leads to a second W-E theorem: W12 = (K2 + U2) -(K1 + U1)
where W12 refers ONLY to those forces for which U's cannot be defined.
One can then consider collections of particles. This introduces
"Internal Forces" (those between the particles) which obey Newton's 3rd Law.
It is at this point that we get into "pseudo-work energy", ie for the whole
collection (Fext)(Scm) = Kcm2 - Kcm1, and into rotation.
Some of the problems that arise at this point are due to treating the
collection of particles as a rigid body.