Every train company has different definitions *and price for trains. Even the English translation is different depending on company. JR calls a 特急 train LIMITED EXPRESS but Keio calls it a SPECIAL EXPRESS.

JR for example has these categories:
Local, Rapid, Express, Limited Express, Super Express (Shinkansen)
Local and Rapid trains do not require extra fee, but express and above require an extra fee because those trains skip more stations and thus faster.
Express trains often have more comfortable reserved seating options, and onboard amenities.

Keio on the other hand has many trains as indicated by shinjukuflyer but everything from Local to Special Express has the same cost and uses the same type of trains (no reserved seating etc). The difference in category just delineates which trains stop and skip at what station, with local trains stopping at all stations, and special expresses stopping at only the major stations.

As for commuter and intercity trains, for Japan, many train companies, especially in the urban areas cooperate with each other's networks that there isn't a clear distinction. For example, some Keio Line trains become the Toei Subway line at Shinjuku(Toei) station, the driver's of each company salute each other before handing off the train in a quick transition.

The best thing is to look at the train map for each company and figure out which trains stop at which station.