How bad are non planar faces?

On the one hand I have learnt, that non planar faces should be avoided, when modelling 3D stuff.

On the other hand I think, they are generated very quickly when manipulating a mesh.

For example a sphere, where one single face is moved, the adjacent faces are turned into non planar faces. See attached file, where the selected face has been moved along the blue arrow.

The lower picture shows the result, after I used the function "triangulate non-planar faces" from menu tools/utilities.

So my question is:
Is it generally recommended, to let Hexagon (or any other modelling tool) let fix non planarity, after some faces or edges of the mesh have been moved?
Or should it only be fixed, if there is a strong amount of non planarity?

Main reason for my question is, that I already encountered some trouble in DAZ Studio and I suppose it was because of these non planar faces. The surface looked kind of strange then in DAZ Studio. But at that time I propably had constructed the mesh in an ugly way. There were quads which were very thin/long, so maybe the problem only arises for these kind of quads in combination with non planarity?