Counting every part of polygons inside another polygon feature class in ArcGIS

I have two polygon feature classes. The first one represents lithostratygraphy distribution. The second one is an artificial grid polygon feature class.

I want to count the absolute number of patches of the lithostratygraphy (not polygons) inside the each mesh of the grid feature class.

1. I use the Intersect tool to divide the classes in the grids.
2. Using the Joins and Relates > Join tool, I calculate the number of polygons in successive meshes.

The problem concerns parts of the patches belonging to the same polygon (e.g. purple upper and lower patches). I would like them to be treated separately, i.e. the result written in the _Counts attribute would be 4, not 3.

How do I split polygons by geometry or how to generate new general geometry feature class like in GeoMedia?