Is there an official name for a notion of "reusably universal"? There are several different (probably inequivalent) notions of computational universality (see for example the last couple pages of http://www.dna.caltech.edu/~woods/download/WoodsNearyTCS07-DRAFT.pdf) and there is no consensus among experts about which notions are most correct (see for example http://cs.nyu.edu/pipermail/fom/2007-October/012148.html).

I'm trying to say something about a particular model of biomolecular computation.  I'd like to argue that it's "more universal" or "more usefully universal" than some other models, because you can construct a universal machine that runs a program and then deletes the input at the end and is ready to run another program.  Contrast this to, say, cellular automata, which can emulate any Turing machine, but then at the end of the computation, you've got a final, unchangeable configuration.  To emulate another TM, you need to define a completely separate CA.  So I'd like to say something is "reusably universal" if it behaves like your desktop, not a CA (i.e., can execute multiple programs without needing to recreate the universe).  Has this notion been formalized anywhere?
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...