As with many things, there can be crossover in these terms. It also doesn't help that rail terminology is vastly different between the U.S. and the U.K. as well as English translations from foreign languages.
However, in the U.S., most transportation systems fall under three categories.
The term "heavy rail," as used by the
American Public Transit Association, refers to the heavy passenger loadings of typical rapid transit (generally subway) systems. The term "heavy" signifies that it carries a very large amount of passengers due to its long trains (six to ten cars, usually) and very frequent headways (just a few minutes apart). Heavy rail systems generally have high speeds (70mph is common) and a dedicated (and usually fenced-off) right-of-way.
Light rail is defined by the same organization to refer to the relatively light (compared to heavy rail) passenger loads. Headways are less frequent (15 to 30 minutes at peak service), and trains are shorter (two to four cars). They often also travel slower and can travel at-grade and even in the street itself. The terms "heavy" and "light" have little, if anything, to do with the weight of the vehicles themselves--light rail cars can be heavier than heavy rail cars.
Commuter rail is considered separate from the above-mentioned modes of transit. Commuter rail (also referred to as regional or suburban rail) typically uses equipment compatible with and following the safety standards of the Federal Railroad Administration and American Association of Railroads. It would normally be possible (though there are exceptions) for commuter rail equipment to run in line with something like a standard Amtrak train or be hooked up to a freight train. Also, outside of the Northeast Corridor, most commuter rail is not electrified, whereas heavy and light rail systems usually use either third-rail or overhead catenary electric power.
The Utah Front Runner uses equipment compatible with and following the standards of the FRA and AAR (in their case, MPI engines and Bombardier Bi-Level coaches). Their employment standards and hours of service rules are governed by the FRA and the applicable Code of Federal Regulations sections. The train can (even if it doesn't normally) run in-line with any freight trains without any special time or distance separation (unlike the San Diego Trolley, where freight trains can use the Trolley's trackage at night after the Trolley has ceased operations). Therefore, it would be considered "commuter rail," even if any crossovers connecting the Front Runner's tracks with the parallel UP tracks were cut.
As an example of how things can appear to cross over terminology, the Siemens Diesel Multiple Units used in the Oceanside-Escondido Sprinter service are one of the rare special cases where it's hard to pinpoint exactly what it is. It's diesel, suburban/intercity, and operates on standard-gauge at-grade trackage with grade crossings, so it would almost seem to be commuter rail. However, the Desiro-class units are not compatible with FRA safety specs and possibly (though I do not know this for sure) lack standard AAR couplers and air brake glad hands to connect to mainline railroad equipment. Therefore, I would classify it as a light-rail system (light rail systems can have grade crossings with normal-looking crossing gates--the nearby San Diego Trolley system as well as the Los Angeles MTA's Blue and Gold lines have grade crossings).
The San Francisco BART is another one that could almost be called commuter rail, since it travels through extensive suburbs. However, the dedicated right-of-way (either underground, elevated, or fenced off), electric propulsion, and nonstandard equipment (wide-gauge and definitely not compatible with AAR standards!) mark it as rapid transit heavy rail. The proposed eBART extension past Pittsburg-Bay Point would likely be something similar to the Sprinter service mentioned above.
A third confusing case would be the Los Angeles MTA Green Line. It has a dedicated ROW primarily in the median of Interstate 105, so it would seem to be heavy rail. However, its top speed is a pokey 55mph, it has less-frequent headways (15 minutes at peak service hours), and runs with shorter trains (two cars). The equipment is more similar to that used on the Gold Line and Blue Line, and so I would classify that as light rail.
But for the most part, all of the transportation systems I can think of in the U.S. (and Canada, for that matter) fall pretty neatly into one of the three above categories. Here are some samplings (though by no means an exhaustive list):
Heavy rail:
Los Angeles MTA Red Line
Boston T Red, Orange, and Blue lines
New York Subway (virtually the entire system)
Washington DC Metro
Atlanta MARTA
San Francisco BART
Chicago RTA El
Light rail:
Los Angeles MTA Green, Blue, and Gold lines
San Diego Trolley
San Jose VTA
San Francisco MUNI
Portland MAX
Seattle South Lake Union Trolley (er, Streetcar--ride it!
)
Seattle Link Light Rail (it's even in the name!)
Boston T Green Line
St. Louis MetroLink
Commuter rail:
SCRRA Metrolink
San Francisco Caltrain
Chicago METRA
Boston MTBA Commuter Rail
New York MTA Metro North
New York MTA Long Island Rail Road
New Jersey Transit (most lines--they operate a light rail/streetcar line somewhere, IIRC)
Maryland MARC
Virginia VRE
New Mexico Rail Runner
Utah Front Runner
