In the piping fields I work in line pipe is used when speaking of API piping used in transmission and distribution lines such as cross country underground pipe.
Process piping is speaking of piping used in a plant to carry a product, such as in a chemical plant. ASTM material is most generally used for process piping, though on occassions API pipe may be used.
Something that is happening quit regularly these days is the gathering (or stealing from landowners) of methane gas, cleaning, odorizing, mixing, & compressing it to a suitable pressure, then distributing it to a gas line to be sold. The piping within the compressor station would fall under the ASME B31.3 piping code and be considered process piping. The piping outside the compressor station would fall under B31.8 and would quite often be called line pipe.