For this shape, the number of intersections is 4, and the number of enclosed areas/regions is five.
For this shape, the number of intersections is 6, and the number of enclosed areas is 7.
It seems reasonable to conclude that number of enclosed areas equals the number of intersections plus one, but I was wondering if this is always the case.
No. Consider $k$ non-intersecting concentric polygons. The number of intersections is zero, but the number of enclosed areas is $k$.
Other violations of the proposed rule occur in the degenerate case with more than two lines intersecting at a point as well as the wholly pathological case of overlapping sides when the number of intersections is infinite.