Consider a family of elliptic curves with some nodal fibers, say $y^2=x(x-1)(x-\lambda)$. Base change to a base where there are two distinct sections that do not intersect the nodes, and glue the sections together. One now has a family of curves, most of which have a node singularity. Where the two sections intersect, there will be a cusp singularity, and where there was previously a node, there are now two nodes.

I think both kinds of bad fibers count as bad reduction.

My argument for the second kind is as follows: Looking at things locally in a neighborhood of the node, the other singularity is invisible, so this might as well be the original, pre-gluing family, in which the nodal singularities were of course bad fibers, as the generic fiber was smooth and they weren't. I do not think that creating a singularity elsewhere on the variety should take a bad fiber and make it good.

Furthermore, note that both kinds of fibers have vanishing cycles.

(Alternately, take an elliptic curve over a number field with positive rank and one semistable prime. Glue a power of an element of infinite order to the zero section, giving the variety a node singularity and creating a prime with two node singularities. This prime should remain a prime of bad reduction.)

Thus I propose the following notion of a fiber of bad reduction: For each singularity on the generic fiber (or on the variety over $K$), its closure is an irreducible closed subset of the total space (or the variety extended to $\mathcal O_K$). Since the singular locus is closed, these closed subsets are entirely singular - they are the "unavoidable" singularities. If a special fiber has an "avoidable" singularity that does not lie on any of these irreducible closed subsets, or it has a singularity on one of these subsets that has a worse singularity type in some meaningful sense than the general one, then it is a fiber of bad reduction. All other fibers are fibers of good reduction. (Similarly, for each singularity in the variety over a number field, we may get as a consequence one or more singularities in its reduction mod $p$. Singularities that are not a consequence of a characteristic zero singularity, or singularities that are worse than the characteristic zero one necessitates, make the prime bad reduction. If that does not occur, it's good reduction.)

As long as the set of points with a singularity type worse than a given singularity type is closed, which I believe it is for all reasonable measurements of singularity badness, then the fibers of good reduction are a nonempty open set. This is clear because the set of bad fibers is a finite union of closed sets that do not contain the generic point.

I do not know if there is a definition of singularity type badness that will work here so that all points with vanishing cycles / ramified Galois representations are fibers of bad reduction.