Earlier quoted context omitted.
Let's work through an example. You are in level, trimmed, manual flight and the trim system suddenly starts pitching the nose down. As a consequence, the aircraft will begin descending and accelerating. Your first response, even before you have recognized that you have a trim runaway situation rather than just some turbulence, is to pull back, to restore level flight level and bring the speed down to where it is supp…
Unfortunately I think you may be misinterpreting that section in the manual. From your senario when the trim moves nose down. You apply nose up on the elevator which makes the trailing edge of the elevator go up which applies a downwards force on the trailing edge of the stabilizer. In order to move the trim back towards neural you need to lift the trailing edge against this force. If the trim is extreme you may have…
Perhaps the phrase "aerodynamically relieve the air loads" means to minimize the aerodynamic force generated by the stabilizer? The stabilizer of a conventional airplane will invariably generate a downwards force when the aircraft is in pitch equilibrium at high speed, but it may be upwards at low speed, even with the C of G within limits. In such cases, the speed at which it has no load will be an intermediate one, and therefore could be either above or below the speed to which the airplane is currently (mis-)trimmed.
The quoted passage says it is referring to correcting what it calls a mis-trim, so if your interpretation is correct, then where it says "in-trim speed", it actually means "the speed to which the airplane is mis-trimmed", while the general usage of the term "in trim" means trimmed to fly at the intended speed, and I would take the phrase "in-trim speed" to refer to that speed - the speed that the pilots intend to fly at, or, in other words, the speed that this out-of-trim airplane would tend to fly at when put back in trim.
Furthermore, if things are as you describe, then the rule could have been more simply and clearly phrased as "reduce your pressure on the control yoke". Maybe your interpretation is correct, but if so, it seems to me to be an unnecessarily confusing way to describe it.