Or is it strictly correct to say these are separate elements, and you can't call the whole a "stabilizer"?
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2 Answers
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If we are talking about the surface area, then yes: the moveable control surface area is defined as being part of the stabiliser area. For instance from the above definition, derived from NACA TN 775.
But indeed, the control surfaces do move, and deflecting them would alter the projected area. Similar to the lift equation, the control surface deflection is considered to generate a change in curvature of the surface, and therefore a change in $C_L$ and $C_M$ of the surface, and not a change in stabiliser area.
$$L_{CS} = C_{L_{CS}} \cdot \frac{1}{2} \rho V^2 \cdot S_{CS}$$
Index $_{CS}$ for Control Surface, with $L_{CS}$ and $C_{L_{CS}}$ changing as the surface deflects, $S_{CS}$ remaining constant as per the image above.
If we're not talking about area, it is indeed obvious that the control surfaces are not part of the fixed stabilisers - because control surfaces can move.
----Update----
NACA TN 775 dates from 1940, when a majority of aeroplanes were manually controlled. If the control surface can indeed "flap in the wind" as stated by @JohnK in his comment, how can it be that the moveable control surface area was included in the stabiliser area?
The answer is that is is more complicated, as commented by @slebetman. Stability considerations for reversible flight controls do consider stick-free static and dynamic stability. During changes in airflow angle and pitch/roll/yaw rate, the airforces exert a hinge moment on the control surface. The demand for stability is now that the change in hinge moment may not result in too much of a control surface deflection.
The following factors determine this:
Friction. There is friction in the cable loop from stick to control surface, which prevents the surface from deflecting until the friction level is exceeded. In large aeroplanes the control loop friction can be considerable.
Mechanical springs. As mentioned in @JohnK's answer, a mechanical spring can be added.
Shift of the surface axis. Moving backwards the axis around which the surface hinges, results in part of the airstream wanting to re-align the control surface.
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1$\begingroup$ There are additional nuances. If the control surface is free to trail because it has no trimming device to fix its deflection angle like a tab or bungee, like the elevator on a C-180/185 for example, it doesn't really contribute to stability in the stick free case; it just trails like a piece of laundry flapping in the breeze. Same with the rudder. Cessna adds bungee springs to the rudders of 180/185s to add yaw stability on floats without having to add ventral fins, by partially "fixing" the rudder (within the bungee spring strength) making it contribute to fixed fin's weathervaning ability. $\endgroup$– John KAug 29, 2021 at 15:07
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1$\begingroup$ @JohnK Not quite true. The free-to-trail part still contributes surface area when it comes to calculating stability (static margin) of the CG $\endgroup$ Aug 30, 2021 at 1:13
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The control surfaces and the stabilizers are not the same. The free dictionary:
Horizontal stabilizer — The horizontal airfoil of an aircraft's tail assembly that is fixed and to which the elevator is hinged
To refer to the assembly, you may use horizontal / vertical empennage, empennage alone meaning both horizontal and vertical surfaces, synonymous of tail.
Stabilizers are stationary parts. Their role is to increase aircraft stability, that is the tendency to return to a stable attitude after a disturbance.
Movable parts, rudder and elevators, are control surfaces, they change the empennage camber. They are used to temporarily set a new attitude. When returned in the neutral position, stabilizers restore the stable attitude.
Empennage components, source
The third component, the tab, is used to trim the movable surface.
There are solutions other than the assembly stabilizer + control surface:
A stabilator (stabilizer + elevator) is a single movable horizontal plane turning around a median point. Therefore there is little resistance when adjusting it as a control surface. Some artificial feedback is added so that the pilot can feel a resistance proportional to the rate of deflection. See this question.
A trimmable horizontal stabiliser (THS) is a classic two-part horizontal assembly but where both parts are movable independantly. It allows to manage stability within an extended range of airspeeds without drastic changes in the overall camber.
The control surface is under pilot control, the movable stabilizer can be under control of computers managing efficiency, stability and flight envelope or under pilot control via the trim wheel (thanks to @Bianfable for adding this information).
When managed by computers it may return to a mechanical linkage when something is wrong with the computers (see Airbus flight control laws).
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1$\begingroup$ Quite a number of GA planes, for instance the Piper Cherokee, have a stabilator. I don't know of any planes that have a rudder/vertical stabilizer designed on the same principle, but I don't know why it wouldn't be possible. $\endgroup$– jamesqfAug 29, 2021 at 4:18
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$\begingroup$ What is that "median point"? The choice of the rotation axis for a stabilator can be driven by many considerations, and not always does it have "little resistance". Remember it is most common in supersonic aircraft, not in GA. $\endgroup$– ZeusAug 30, 2021 at 0:40
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$\begingroup$ @Zeus: This is detailed in Axis of rotation of tail surfaces in particular your answer deals with Mach tuck and swept airfoils. $\endgroup$– minsAug 30, 2021 at 7:57