The 'topological' in topological order means 'robust against ANY local perturbations'.
According to such a definition, topological insulator is not 'topological' since its properties are not robust against ANY local perturbations, such as the perturbation that break the U(1) and time reversal symmetry. So a more proper name for
topological insulator is 'U(1) and time-reversal symmetry protected insulator', which is one example of SPT order.
Some example of topologically ordered states (in the sense of 'robust against ANY local perturbations'):
1) $\nu=\frac{1}{3}$ FQH state
2) $Z_2$ spin liquid state
3) $\nu=1$ IQH state
4) $E_8$ bosonic QH state
The example 3) and 4) have no non-trivial topological quasi-particles (ie no non-trivial statistics, no non-trivial topological degeneracy), but have
gapless edge state that is 'robust against ANY local perturbations'.
-- Edit -- (I lifted some discussions below to here):
There are two kinds of topology in math. The "topology" in "topological order" is directly related to the first kind of topology in mathematics, as in algebraic topology, homology, cohomology, tensor category. The "topology"
in "topological order"
is different from the "topology"
in "topological insulator". The "topology"
in "topological insulator"
is related to the second kind of topology in mathematics, as in mapping class, homotopy, K-theory, etc. The first kind of topology is algebraic, while the second kind of topology is related to the continuous manifold of finite dimensions. We may also say that the first kind of topology is "quantum", while the second kind of topology is "classical".
The correct way to describe any gapped phases (such as topological orders and topological insulators) is to use the first kind of topology -- "quantum" topology, because the gapped phases are usually interacting. The second kind of topology -- the "classical" topology -- can be used to describe the one-body physics (include free fermion systems). The "classical" topology cannot be used to describe interacting many-body systems, which need "quantum topology".
One needs to go beyond "filling energy level" picture to understand topological order (the first kind of topology). Our education in traditional condensed matter physics (or traditional many-body physics) is almost all about "filling energy levels" (such as Landau Fermi liquid theory, band theory, etc), which is a trap that limit our imagination. The second kind of topology (the "topology"
in "topological insulator") can be understood within the framework of "filling energy level" picture.
To answer the question What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this? I like to say that topological order is algebraic, not geometric. So the topological invariants of topological order are very different from Chern numbers. The robust ground state degenercy and the robust non-Abelian geometric phases of the degenerate ground states are the topological invariants of topological order (which are the analogues of the Chern number).
Let me first answer your question "is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?"
(1) Topological superconductors, by definition, are free fermion states that have time-reversal symmetry but no U(1) symmetry (just like topological insulator always have time-reversal and U(1) symmetries by definition). Topological superconductor are not p+ip superconductors in 2+1D. But it can be p-wave superconductors in 1+1D.
(2) 1+1D topological superconductor is a SET state with a Majorana-zero-mode at the chain end. But time reversal symmetry is not important. Even if we break the time reversal symmetry, the Majorana-zero-mode still appear at chain end. In higher dimensions, topological superconductors have no topological order. So they cannot be SET states.
(3) In higher dimensions, topological superconductors are SPT states.
The terminology is very confusing in literature:
(1) Topological insulator has trivial topological order, while topological superconductors have topological order in 1+1D and no topological order in higher dimensions.
(2) 3+1D s-wave superconductors (or text-book s-wave superconductors which do not have dynamical U(1) gauge field) have no topological order, while 3+1D real-life s-wave superconductors with dynamical U(1) gauge field have a Z2 topological order. So 3+1D real-life topological superconductors (with dynamical U(1) gauge field and time reversal symmetry) are SET states.
(3) p+ip BCS superconductor in 2+1D (without dynamical U(1) gauge field) has a non-trivial topological order (ie LRE) as defined by local unitary (LU) transformations. Even nu=1 IQH state has a non-trivial topological order (LRE) as defined by LU transformations. Majorana chain is also LRE (ie topologically ordered). Kitaev does not use LU transformation to define LRE, which leads to different definition of LRE.
Best Answer
I think the first helpful fact to clarify is that there are two different kinds of topological phases: there are so-called Symmetry Protected Topological (SPT) Phases (displaying 'symmetry protected topological order') and there are (intrinsic) Topological Phases (displaying '(intrinsic) topological order'). As some quick examples: topological insulators and topological superconductors are example of the former, whereas the quantum Hall states and various spin liquids are examples of the latter.
There are no clear-cut one-line definitions of either that everyone would agree with, but both have clear characteristics. But let me first make a different point, to clarify both the similarity and difference with the more usual spontaneous symmetry breaking phases: all phases characterized by spontaneous symmetry breaking are also symmetry protected! Take for example the 1D quantum Ising model $$H = \sum_n (- \sigma^z_n \sigma^z_{n+1} + g \sigma^x_n)$$ This spin hamiltonian has a $\mathbb Z_2$ symmetry defined by flipping the spin in the $z$-direction, and as is well-known the ground state spontaneously breaks this symmetry for $g <1$, in this case called a ferromagnetic phase. Now, this phase is stable under arbitrary perturbations, on the condition that these perturbations respect this $\mathbb Z_2$ symmetry. As soon as you add for example a term $h \sigma^z$, then the ferromagnetic phase is no longer well-defined, or put differently: it is no longer protected, meaning you can continuously deform the Hamiltonian and ground state to become a trivial paramagnet without encountering any phase transitions. So in a sense whenever we have spontaneous symmetry breaking, we should call it Symmetry Protected Spontaneous Symmetry Breaking. That would however be a redundancy in description, since any spontaneous symmetry breaking needs such symmetry protection.
That is different for topological phases: some phases are, as you say, stable under arbitrary perturbations. That is, you cannot connect the phase to a trivial(*) phase without encountering a phase transition. These are called (intrinsic) topological phases. So this is a whole different game, and in a way it is remarkable that such phases exist at all. The intuitive reason is that these phases have long range entanglement in the ground state (meaning that far-away sites of the system can be strongly entangled with one another), and you cannot undo such entanglement by mere local perturbations. This long range entanglement is also the cause for many interesting physical consequences. In particular such intrinsic topological phases have very interesting low energy excitations (the concept of 'fractionalization' comes up here: at low energy the system behaves as if there are exotic particles, more complex than the actual particles that make up the system on a more fundamental level). There are two conceptual reasons for why such phases are called 'topological': (1) in the mathematical treatment of these phases, the mathematical field of topology often pops up (for example in the context of the low energy theory being a topological quantum field theory, or by the order parameter of the phase being a certain topological invariant); (2) on a more mundane level, the system behaves differently if you put it on spaces with different topology: the system for example behaves differently if you put it on a open slab (e.g. a rectangle) versus on at torus (i.e. no boundaries). Both (1) and (2) are associated to the fact that such topological phases have interesting phenomena on their physical boundaries.
The previous paragraph focused more on such phases with intrinsic topological order, where symmetry is not that important (although there are interesting formulations of topological order where symmetry comes up, but that would be too much of a digression). As mentioned before, there are also symmetry protected topological phases: these have more short range entanglement, and like symmetry breaking phases the phases are only well-defined if the Hamiltonian has a certain symmetry. However the ground state does not break this symmetry spontaneously. Instead the system behaves 'topologically' in the above sense: often topology comes up in its mathematical description, and there peculiar effects going on around the boundary (like the gapless edges of topological insulators). The main physical difference between the intrinsic and the symmetry protected topological phases, is that the former hosts fascinating behaviour in the bulk (i.e. away from the boundaries), in the form of exotic/fractional quasi-particles, whereas the latter (being the symmetry protected topological phases) doesn't have much going on in the bulk, and the mainly interesting physics is on the boundary (however it does have some interesting structure in the entanglement of the bulk --despite the entanglement being short range-- but this is quite difficult to measure physically).
I hope this gives a bit of the general overview that you were asking for. Let me know if I can clarify further.
(*) One can wonder how one defines the notion of 'trivial phase'. A good rule of thumb is: a phase is trivial if one cannot say anything interesting about it.