If you have a typical precision positioning application, then the DC brushless servo motor is a logical choice. Brushed motors should not be used due to the problems associated with brush wear and heat generated at the interface. These are not a common design in linear servos.
Stepper motors cannot be tuned to the load or position-feedback easily applied. This limits them to very light loads, usually less than 20 lbs. Because of the large coil weights of linear steppers, end of move settling can also be problematic. Linear steppers are limited in speed, limiting machine throughput.
The requirements that usually point to a linear motor are long life or number of cycles, high speed, very precise positioning, no wear, low noise, clean room or vacuum compatibility, and low maintenance.
Induction motors do not use magnets but are not as efficient as brushless motors. Induction motors are also very expensive, typically used in extremely high acceleration applications such as roller coasters and require very high powered amplifiers.