Is there an official name for a notion of "reusably universal"?
There has been work in the PL/systems community on the semantics and modelling of operating systems.  As you point out, what you are talking about is much like an OS: it does something, but is guaranteed (well, in the case of an OS, guaranteed-ish) to return to a "ground state."  The PL people may not have formalized your notion of reusably universal, but you might find some inspiration there.

Your formalization will have to capture the difference between "a universal machine that, after it runs with one input, if you replace the input with another input it is ready to go" and "a universal machine that, given a sequence of input programs, runs them in succession."  And of course, all reasonable notions of universal machine probably satisfy the latter requirement.  So it seems quite tricky...