Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data?
Motivation and attempt at an answer: The question should be local, so we can base change to get $Y/\mathbb{Q}_p$. Good reduction should mean something like: There exists a regular, proper $\mathcal{Y}/\mathbb{Z}_p$ such that the generic fiber is isomorphic to $Y$ and the special fiber is smooth. This is potentially a bad definition:
This behavior for curves of genus $g\geq 1$ is nice, because in the appropriate category of such models there is a partial order by dominating. One can prove that there is a unique, regular, proper, minimal model of the curve which can be used to determine reduction type.
(Edited paragraph from comments) Note that given such a minimal model, one can blow up points on the special fiber. These blowups are still generically isomorphisms and hence models. They are no longer minimal, though.
Related question 1: For higher dimensional varieties, if you have $2$ minimal, regular, proper models (these may not be unique), if one has a nonsingular special fiber, then must the other as well? This would give a well-defined way to determine reduction type.
Question 2: Is this approach just overly complicated (i.e. has this theory been worked out in some other way)?