I believe that K50 and K51 are the internal BMW model designations for the GS LC and GSA LC respectively.
So for example, if you see panniers being advertised as suitable for the K50 then that means they are designed for the LC version of the GS as pre-LC models have the exhaust and shaft drive on the opposite sides which obviously affects the design of the panniers. It also means that if they are the BMW Vario cases, then they will require the GS LC version of the passenger footpeg hangers because these provide a prong which acts as the front pannier mount, and will therefore not mount on the GSA which has different footrest hangers with fixing brackets for the tubular pannier frames which fit the GSA.