"Exclusive scattering" means that all final particles are known. They may be all detected directly, or possibly one may be inferred from conservation laws. For instance in exclusive pi+ electroproduction :
electron + proton -> electron + neutron + positive pion
one may detect the neutron directly, or infer its presence by conservation of electric charge, and the missing mass of the system (electron + positive pion).
Now usually when we use "inclusive" we suppose that only one particle is detected. For instance only the electron in
(I) electron + proton -> electron + X
then X can be anything with the right quantum numbers. It is very useful because by unitarity (optical theorem) the cross-section for the process (I) is equal to the (imaginary part of the) amplitude for forward scattering where the initial state is the same as the final state (in momentum space).
There are in addition semi-inclusive processes where two or more particles are detected, for instance
electron + proton -> electron + positive pion + X
