The "other requirements" are the real requirements, the string business is just a gloss, which omits the most important S-matrix assumptions.. There is another question regarding this here: Are There Strings that aren't Chew-ish? (and the linked discussion explains some of the issues).
String theory is not just a theory of strings. In its simplest formulation, it is a theory of strings which can only interact by exchanging other strings, not by exchanging point particles (or anything else). This excludes things like atomic polymers, or strings made out of points, or strings that interact by self-intersection, except to the extent that you can view the special string-theory strings as made out of string bits, like in Matrix theory.
The historical marginalization of S-matrix theory and its practitioners is the main reason that the string assumption is played up, and the bootstrap assumptions are played down. Bootstrap was politically unpopular, and any theory that said "bootstrap" would be ignored in the 1980s and 1990s.
A good list of requirements on string theory is this:
- There are strings, so the spectrum of the theory is the oscillation spectrum of a string worldsheet action.
- the exchange of strings is the source of all the (perturbative) forces in string theory. This means that once you know the string spectrum, you know the interactions are by summing over all intermediate states of the string alone, with nothing else.
- The scattering is Regge-soft scattering in Regge limits. This requirement is technical, and hard to state for a general audience, so it is left out. What it says is that the sum over all intermediate string states gives cancelling amplitude from particles of different spins, and that these cancellations lead to an amplitude that falls off faster than a power at very high energies. although each spin-n state gives an amplitude that blows up ever harder at high energies. This is sometimes called the bootstrap assumption, the assumption that everything in the theory is a bound state which is part of a family of related bound states which together give softer scattering than each one individually.
- The exchange of strings in the t-channel is the same as exchange in the s-channel, and does not require counting the particles separately. This gives the world-sheet picture, from the symmetry properties of tree diagrams in such a theory. That the interactions are by worldsheet just is not derivable from the spectrum alone, without the assumption that the spectrum bootstraps, and doesn't resolve to something else at short distances.
These requirements partially overlap, nobody has made orthogonal axioms. Together with the following rule:
- The string must have a fermionic excitation
They should be enough to uniquely determine the perturbative superstring theories. It is clear that there is at least one string theory that people missed completely, this is Simeon Hellerman's M-theory-on-a-Klein-bottle strings, and there are tons of different vacua which could be thought of as new theories in this formulation, because each S-matrix is a different theory.
These perturbative string theories link up non-perturbatively into an ubertheory called M-theory. A theory today would be called part of string theory if it is a description of some configuration of M-theory. There is no full enumeration of these, so it is impossible to say exactly what a configuration is, although there is a partial list, and there is no real arbitrariness.