There are different basic types as well...
First generation jet transports (early models of same...speaking B707 here) had a parallel yaw damper, where the rudder pedals moved continuously as the rate of input from the vertical gyro was received to the yaw damper actuator.
On these early models, it was required to switch the yaw damper OFF for takeoff and landing, because if it was not done, rudder forces for the pilot would be insurmountable in the event of an outboard engine failure...and marginally just possible with an inboard engine failed.
Even tho these aircraft had a hydraulic powered rudder, control forces were heavy with an outboard engine failed at takeoff thrust.
Later models of the 707 had series yaw dampers, and these, due to the much improved design, were full time...ie: not required to be switched off, except if they malfunctioned...and the usual malfunctions were, excessive rate response or rudder hardover...the latter being very bad news, especially just after takeoff.
Second generation jet transports (speaking Lockheed TriStar here) had both types.
The SAS (stability augmentation system) supplied series yaw damping/turn co-ordination...and parallel operation for runway alignment and rollout...the parallel mode used for automatic approach/landings only, with one or two autopilots engaged.