Vessbot - thanks for bringing this topic out of the LA 610 discussion for a deeper dive here. One of the key elements to this topic is the definition of "trim". I see two related, but distinct meanings: (1) move the horizontal tail or what ever surface, tab, etc. is used to allow repositioning of the primary pitch control surface (most often an elevator), and (2) relieve pitch control input forces. On an unaugmented airplane where elevator is controlled by the pilots pitch control input (no Hal involved) these two definitions of "trim" are one and the same. On an airplane that has augmented pitch control they are not.
An example of an airplane with no augmentation of the elevator control is the 737. Making manual "trim" inputs on this model drives the horizontal stabilizer directly. There are functions on the 737 that drive the stabilizer directly (STS and MCAS with autopilot disconnected, and stabilizer offload with autopilot engaged) that can cloud the picture, but in general manual pitch trim on a 737 both moves the stabilizer directly and impact the level of column force that the flight crew must maintain.
An example of an airplane with augmented elevator control is the 777. The 777 elevator responses to the pilot's pitch control input through both a proportional, feed forward path and through feedback paths that take into account pilot maneuver demand and airplane response. Such a system can be designed to reject some pitching moment disturbances such as those resulting from thrust and configuration changes and preserve or even generate others such as providing the desired level of positive speed stability. On 777 this pitch augmentation system is given the name C*U. With C*U a separate, automated function monitors elevator position and automatically commands stabilizer motion to maintain the desired long term, steady state position of the elevator. Steady column forces that the pilot must maintain with C*U are related to maneuvers and changes in speed. With C*U, the pilot's manual pitch trim input does not drive the stabilizer. Pilot manual pitch trim input manages a reference speed that serves as the trim point for hands off flight. In this way, 777 has separate portions of the C*U control law that address the two different meanings of "trim".