What you are running into is the fact that people are using confusingly similar language for rather different concepts. The term "topological materials" in particular seems to be used differently by different people. I advocate an inclusive definition, where "topological materials" encompass two subclasses of materials:
(Time-reversal-symmetric) Topological insulators are the most well-known example of SPT states, and the quantum Hall states the most well-known examples of topological orders. Topologically ordered states are robust to any local perturbations, whereas SPTs are robust only to local perturbations respecting the symmetry.
As for fractional excitations, they can occur only in states with topological orders. In other words, opological insulators do not have fractional excitations. If that's not confusing enough, there are also (interacting) generalizations of topological insulators which have topological order. These are known as fractional topological insulators.
- What are topological excitations?
They are excitations that cannot be created by local operators. They may have fractional statistics and carry fractional quantum numbers, but this is not generally a requirement. Fractional excitations are necessarily topological. For example, you cannot locally create a particle of charge $e/3$ in a system built from electrons, protons, and neutrons. However, many electrons can collectively fractionalize into multiple quasiparticles with fractional charge.
Note that the presence of topological excitations implies that the state has topological order, but the reverse does not hold. Topologically ordered states without topological excitations are said to have invertible topological order. The chief example would be the integer quantum Hall states.
- Is the original statement true? That is, is a material topological if and only if it has fractional excitations?
No, this does not hold even when restricted to states with topological order.
If the material has fractional excitations it has topological order, and is thus characterized by its topological ground state degeneracy, or long-range entanglement. However, there are topologically ordered states without fractional excitations. Examples include the Kitaev Majorana chain or integer quantum Hall states. For more examples see Table I in Xiao-Gang Wen's review Zoo of quantum-topological phases of matter.
By the way, I can warmly recommend that review paper. It cleared up some of my confusions on this topic. It's not the easiest read because it covers a lot of ground, but it's one of the clearest expositions of how different concepts are related, and how different groups have used different terminology and classifications.