To try to help answer the discussion I would add the following:
You should only really consider the stability characteristics of an aircraft when it is in trimmed, balanced flight at constant thrust (and technically constant mass).
Imagine if you will a "traditional" aircraft configuration - with a wing producing the main lift and a horizontal tailplane set at the end of the lever arm to produce the balancing effect (and generate the pitching forces). When the aircraft is all perfectly trimmed out and the speed is pegged, then T=D and L=W. The L in this case is the sum of the wing lift and the tailplane lift.
Now, without changing the thrust and without changing the trim, change the speed slightly by, for example, shoving the stick forward for a couple of seconds and then letting go. The motion of the aircraft will now follow what is known as a phugoid - as the nose has been pushed down the speed will increase. As the speed increases the corresponding increase in lift (wing and tp) will pitch the aircraft back up. It will overshoot the trimmed condition and get slow and pitch down. This cycle will be repeated a number of times depending upon "how" longitudinally stable the aircraft currently is. If it is "highly" stable the number of cycles will be small; if it is only "slightly" stable it will continue this phugoid over many cycles before 'eventually' returning to the trimmed conditions (assuming no other external influences (gusts) and constant mass/thrust).
Now, "what defines how much stability the aircraft currently has?" I hear you ask. Well that is the subject of the original question, but in real terms it is the CG that we have control over rather than the CP. The stability comes from the relative position of the CG wrt the lift arrow of the wing and the lift arrow of the tailplane and relative amount of lift from each. Two up arrows and one down arrow. A see-saw with the CG as the pivot point.
The range of allowable CG positions are determined through flight test. As the CG moves to the boundaries the aircraft becomes less stable. If you take the CG out side the limits the aircraft can become unstable and that is a whole other subject...
Hope that helps, best I can do without using my hands or drawing a picture.
Cheers
Tarnished