What does MULx operation in SNOW 3G correspond to?
It essentially is the value $02 \times V$ in the field $GF(2^8)$, and so MulxPOW(V, I, c) is essentially $02^i \times V$ in the field $GF(2^8)$, where $c$ specifies the representation used.

What is a representation: there are a lot of ways to map between 8 bit values and the members of the abstract $GF(2^8)$ field, and a number of those mappings are plausible (unless $GF(p)$, which has an obvious mapping between integers in the range $[0, p-1]$ and members of that field - so obvious that we typically ignore it).

What Snow does is use several different mappings (representations) for the various SBoxes, by specifying different $c$ values for the different sboxes.