Radial piston motors are usually high torque low speed motors where the piston are placed at 90 degrees to the rotational movement of the drive shaft. They are used for some mini and midi excavator swing functions as they don't necessarily need a reduction gear. These motors are very rugged and expensive. They do not handle contamination very well. These motors can be built to be bi-directional. I've seen radial piston motors running in systems up to 5,000 PSI.
Axial piston motors set the pistons parallel to the drive shaft and obtain rotational movement by forcing the piston feet against a tilted plate usually known as a swash plate. These motors are usually high speed and low torque and are commonly found with gear reduction units mechanically attached to the output shafts. These types of motors are built to very close tolerances, will be damaged easily by contaminated oil and are very expensive. These motors can be built to be bi-directional. Axial piston pumps and motors are common in 5,000 PSI systems.
Gear motors are exactly as described. They are two gear that mesh with each other. Oil is forced in one side of the pump and around the outside of the gears in the spaces between the teeth. These motors can be built bi-directional and they are very common throughout most industries. They are rugged and can take more contamination than either of the piston motors above. They are relatively inexpensive compared to piston type motors. These types of motors can be found in systems running up to around 2,900 PSI. Pump cases are usually the weak point and crack or just start leaking after a finite amount of pressure cycles.
The gerotor pump is a single gear with external teeth meshing inside a circle gear that only meshes in one spot with the internal gear. They are inexpensive and mostly thought of in the hydraulics world as cheap. They have low torque and usually low to mid speed. The type is used more for lube oil pumps in many internal combustion engines and are common as power steering pumps in cars. They will take some contamination and lose efficiency but still provide enough flow to accomplish their task. These types of motors are low pressure, low to medium flow, and are the least expensive to purchase.
Your sales rep has a point about contamination being introduced through the quick couplers but that shouldn't be much of a problem with good cleaning practices when installing the cutter on the machine. On the other hand a gear motor is a whole lot less expensive to replace compared to a piston motor and shock loads on cutter heads clearing brush are very common occurrences. There are other here that are more experienced with mulchers and cutter heads that can comment on which type of drive motor works better for them.