I have a question about the definition of an elliptic surface. One defines an elliptic surface $S$ over a base curve $C$ (over some field $k$) as a surjective morphism $f: S \to C$ such that almost all fibres are smooth genus 1 curves, which is moreover relatively minimal with respect to the elliptic fibration - i.e. no fibre contains a smooth rational curve with self-intersection $-1$ ("exceptional curve of the first kind").

Often one also includes the condition that there is at least one singular fibre.

I've never understood why one really wants to include this last condition. For example, there exist "elliptic surfaces without singular fibre" which are NOT isomorphic to a trivial surface, i.e. a product of $C$ with another curve. (I'm not sure whether this can happen for $C = \mathbb{P}^1$ though.)

Could someone shed some light on this matter?