Introduction
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TiO$_2$ has an equilibrium concentration $c_{eq}$ of all symmetry-unique oxygen vacancies, which is given by
$c_{eq} = \frac{N_{O_i}}{N_O} = \exp \left( -\frac{\Delta G_{vac}}{k_B T} \right)$
where $N_{O_i}$ is the number of oxygen ($O$) of symmetry type $i$ in a sample or simulation cell of TiO$_2$, $N_O$ is the total number of oxygen in the same sample or simulation cell, $k_B$ is the Boltzmann constant, and $T$ is the temperature.
Note that this equation assumes that the oxygen vacancies do not interact, which allows the use of the mixing entropy of an ideal solution. This equation also assumes the formation of a neutral oxygen vacancy, i.e., O$^{2-}$ donates two electrons to TiO$_2$, which usually reduces two neighboring Ti$^{4+}$ to Ti$^{3+}$, as it leaves to form half a mole of O$_2$ gas. Since TiO$_2$ has a bandgap, it also can form charged vacancies (you can find an excellent and very recent summary of this here, https://doi.org/10.1038/s41578-022-00433-0).
$\Delta G_{vac}$ is the Gibbs free energy ($G$) of oxygen vacancy ($vac$) formation, which contains the O$_2$ gas pressure dependence and the effect of changes in the vibrational entropy.
The answer
Question 1: is it reasonable to decide it by choosing the configuration with the lowest energy?"
If one oxygen vacancy configuration has a much lower formation energy than the others, probably.
Question 2: What is the rule for deciding the position of defects?
All defect positions (i.e., symmetry-unique sites) have an equilibrium concentration...some larger than others...some experimentally detectable, others not. The "rules" deciding their concentrations are the first and second laws of thermodynamics, which give the equation for $c_{eq}$ above.
it doesn't feel quite right to always choose the configuration with the lowest energy, because it means this configuration is quite stable, if I wanna use it as a catalyst, I might want it to be active.
Unfortunately, we don't get to choose the lowest energy – or, better yet, highest concentration – defect configuration. This is controlled by crystal bond strengths (O$^{2-}$–Ti$^{4+}$), cation reducibility (Ti$^{4+} \rightarrow$ Ti$^{3+}$), the bandgap, polymorph stability (Rutile vs. Anatase vs. Brookite), etc. (Shameless plug: To learn more about the factors controlling neutral oxygen vacancy formation in metal oxides, check out https://doi.org/10.1021/jacs.1c05570)
I agree that sometimes a material isn't active enough (i.e., it doesn't form oxygen vacancies easily enough) to be a catalyst. First, you need to identify what is active enough, i.e., an acceptable range for the oxygen vacancy formation energy for your catalysis application; this is a prerequisite for catalyst design! Then, you can search for either (1) bulk materials with good oxygen vacancy formation energies or (2) dopants that tune a material's oxygen vacancy formation energies to within the acceptable range.
I hope this is helpful!
