I'll offer a simpler and more direct answer: in a trimmed condition, it is a total CP that must coincide with the CG, and this is essentially by definition, and it doesn't "depend on the aircraft in question".
(Here for simplicity we restrict ourselves to the pitch motion and ignore possible effect of thrust and drag, which line may not exactly pass through CG and which moment will then need to be compensated).
In the same condition, AC will be behind CG (and CP) for a statically stable (in pitch) airplane. Imagine the airplane is disturbed and pitches up (or experiences an updraft; the fact is, its AoA temporarily increases). The extra lift due to increased AoA is applied at AC (now by the definition of AC), and since AC is behind CG, it creates a pitch-down moment which returns the aircraft to the original AoA until this extra lift is eliminated and everything returns to the former balance. This is the definition of (static) stability.(*)
From this it follows - and it is important to realise it clearly - that an airplane is trimmed for a certain AoA. Not the airspeed, not the pitch. At a given trim setting (for a steady level flight), you can fly at a higher speed and higher load, for example, in a turn or spiral.
Another thing that may help to avoid confusion is to understand that AC is a very theoretical, abstract point. It is defined purely for the convenience of stability analysis, and defined such that it doesn't move (within reasonable AoA). So in flight you can't "position" it at will, just as well as in most cases you can't move CG much. In a sense, all control is done by shifting the CP (of the whole aircraft).
At the same time, CP and CG can be though of as "real" points where a known real force is applied (although both are also abstractions in reality). When you need a balance, i.e. lack of total moment, you want the lift and gravity to act at the same point. (Remember we neglected moments from other forces, which are often small).
(*) Longitudinal stability is often incorrectly explained through speed: the aircraft pitches up, loses speed, then "wants" to return to the trimmed speed by pitching down to accelerate. This is wrong; the pitch-down moment arises immediately as the AoA grows, much sooner than any appreciable change of speed (if any) happens. When flight dynamicists speak of longitudinal static stability - and that's exactly where the concept of AC appears - they really speak about AoA stability. Airspeed is not even a factor there (or rather, change of airspeed is not). It is this AoA stability that makes the aircraft flyable by humans.
When we disturb only airspeed, as in Carlo's answer, e.g. by increasing thrust, a different process is involved. First (ignoring some fine effects), lift starts to increase quickly (as square of speed). But this increase doesn't come at AC; remember AC is only about AoA! Because we maintain the same AoA (at least initially), you get proportional increase of lift on the wing and tail so that the total balance remains, and the lift increases at CP=CG. As a result, the airplane starts to accelerate upwards (but not 'climb' in a normal sense). Now this means decrease of AoA, and, apart from damping of lift itself, this triggers the normal AoA response, that is, the attempt to increase it back to the trimmed AoA, i.e. pitch up.
Note I didn't mention the tail downforce. It's not really a requirement. It's just an exaggerated way to ensure a rearward AC position. But you don't need to involve it to explain stability if you already defined AC. The downforce is just an implementation detail, as programmers say.
