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The way I understand the situation is what is typically/usually
called the 'internal energy' in thermodynamics/statistical physics
is simply the total (plain old as JD says) energy but *as seen* in a
particular frame in which the system's center-of-mass is at rest
and not rotating, or, more specifically, the frame is one in which
the system's total momentum is zero and the total angular momentum
about its CM is zero.
Such a frame may be either inertial or non-inertial
in any inertial frame it possesses a number of extensive constants of
motion stemming from a combination of Noether's Theorem and Poincare
invariance.
If the system is not fully isolated but is in weak interaction with
an environment of a number of fixed intensive potentials, [....]
Another way to keep a system at rest for which the use of a container
is not appropriate (e.g. an expanding gas cloud) is to rewrite the
system's Hamiltonian in a coordinate system that treats the CM
degrees of freedom as their own dynamical variables, and treat the
rest of the degrees of freedom in terms of relative coordinates
measured WRT the CM.
It makes the problem analysis simpler.