When you have atoms bonded all in one plane, there will be $p$ orbitals oriented perpendicular to the plane which may not interact significantly with adjacent atoms. Such orbitals would then be called nonbonding.
We may compare water with carbon dioxide. Introductory textbooks often describe the oxygen in water as having a distorted $sp^3$ hybridization, but in reality only two oxygen $p$ orbitals mix with the hydrogen and oxygen $s$ orbitals to form the bonds. The third, perpendicularly oriented $p$ orbital is a nonbonding $p$ orbital. We can see this orbital (HOMO) in the middle of this diagram from flickr.com:
In contrast, carbon dioxide does have the $p$ orbitals on the central carbon atom interacting strongly with out-of-plane orbitals on the oxygen atoms forming the familiar $\pi$ bonds.
Now let's look at titanium dioxide. Titanium dioxide may come in different phases depending on temperature and pressure, but the phase we generally see under ambient conditions is the rutile structure by WP user Ben Mills:
The red atoms are the oxygen atoms in titanium dioxide, and these are seen to have a distorted, but planar, triangular coordination with the gray titanium atoms (an equilateral triangular coordination does not fit with the octahedral coordination of the titanium). The anatase form described in the OP's reference has a more complex atomic arrangement but similar triangular coordination around each oxygen atom. So, there are perpendicularly oriented $p$ orbitals. The titanium atoms also have out-of-plane orbitals, but unlike the carbon in carbon dioxide these do not interact strongly with the out-of-plane oxygen $p$ orbitals. The difference in orbital energy and size is too great. So the out-of-plane oxygen $p$ orbitals are more like their counterparts in water, essentially nonbonding.
