I'm not sure I would say they have similar movement, although some people trying the RH their first time can end up throwing the CK if they fail to rotate the standing leg; and young kids can manage to throw the RH even without any standing foot rotation due to their bodies being more like Gumby.
But both kicks are X-axis kicks - they move from side-to-side, rather than up/down, or away from the body; and so in this way they are similar. But they differ in so many ways, and even a novice to the style would easily discern the kicks are very different.
I would also say that if you looked at only the motion of the foot, the shape it makes could be similar. But even here, things can get different. Suppose you dipped the kicking foot in luminescent paint, then turned off the lights except a black light. You would notice a very similar shape in the resulting blob of light between the kicks. But, if you taped a laser light to the sole of the foot, you would notice that the laser light travels only on the walls (RH); and only briefly on the walls and chiefly on the ceiling (CK - both varieties).
You do need to look at the rest of the body for the difference:
Rotational of standing foot
In the CK, you can get away with little or no rotation of the supporting foot, while in the RH, you must rotate (unless you are said Gumby...)
Shape of the knee
In the CK, your knee remains more or less locked, and the power comes from the whole leg; in the RH, you snap the foot from the shin's extension from the knee.
Kick speed
In measuring a kick's speed, the important speed is the thing that's striking: that would be the foot. The foot, in both kicks, sits on the end of a radius. In the CK, the radius is the whole leg, which moves about the hip. In the RH, there could be two radii. One is always the knee-to-foot radius (basically, the shin), the other is what advanced kickers use and is the moving radius of the thigh (hip-to-knee). A beginner uses the thigh radius to get the knee into position, stops, and then throws the kick (utilizing only one radius), whereas the advanced kicker can combine the independently-moving radii such that the shin radius can whip from the thigh radius.
But the CK only moves from the hip, and to get the foot's speed means to develop the fast twitch muscles of the hips to get that leg to move fast - that is no easy task. As a result, for many kickers, the CK tends to be a much slower kick than the RH
Versatility
The CK can't easily be thrown with a jump (with other knee being lifted first), whereas the RH can easily be done so. Both are easily thrown with a hop (no lifted knee).
The CK kick DOES have versatility: if the timing is right, it can block or parry any kick, whereas a RH is not suited for kick blocking. The CK can be used to off-balance the opponent when parrying, another advantage it has over the RH.
Striking parts
The CK can only land with the inside of the foot, the sole, the heel, or maybe the calf; the RH can only land with the instep, the ball of toes, or the shin.
Repeated kicks
The CK lands with a single kick only; the RH can be initiated with a fake front, RH, or side kick; it can also be thrown and recoiled into chamber and thrown again and again without going through the initial rotation - and do so indefinitely (which is a drill in of itself for this particular variant). The CK must be thrown, and cannot be interrupted, and is very difficult to control, one of the many reasons it has fallen out of favor.
I think you pointed this out in your comment about double-kicks.
Muscle physiology
When throwing the CK, it is necessary to develop the adductor muscles, the inner thigh muscles, because they need to draw the leg from the outside to inside.
When throwing the RH, the adductors are used much less, and the abductors (outer muscles) are used more to open the hips to lift the knee.
Joint physiology
RH kicks can be very dangerous to the knee joint, especially if throwing lots of air kicks and the kicks are fully extended. The CK does not incur the same impact in any of the joints in the leg.
In your commentary, you mention the motion of lifting the knee and extending. Perhaps this is where styles may differ - and I don't mean like ITF / ATA / KukkiWon, etc, I mean like between Taekwondo-in. I don't usually lift the knee in the same way between the kicks. For me, the knee lifts to the outside (for the out-to-in variant of the CK; of course, I think we can all agree that the in-to-out has a very different motion). For the RH, I lift the knee in a direct line to the target, such that the knee points to the target; then the foot flicks out to strike the target.
As to differences in style, I doubt one style differs over the other. I have been kicking the same way for almost 40 years, and have been in and out of both Kukkiwon, Chung Do Kwan, Tang Soo Do, and ITF schools, and not a single one wanted me to change in any significant way. Where my problems lie with the RH is that it's a hip-ripper for me; I've lost flexibility over the years, and I struggle to get the kick out because of the needed hip rotation and the lack of hip flexibility. So my kicks are generally very low compared to the CK. My RH kicks barely make head-high, and when I do, there is less control for me. With the CK, I can easily strike the head, and, if I wanted, I can stop the kick on a dime. (Ok, maybe a quarter...) As a result, I dip my head more with the RH as I try to reach high; that causes the hands to compensate for balance. The whole package looks messy for me when I throw high roundhouses.
I will say that for the RH, my ITF schools like to start beginners out by having their knees take a wide berth to the chamber, while my WTF-oriented schools like a more direct position of the knee. The difference here is in speed. But, WTF schools do tend start their students out pretty much the same as their ITF counterparts, and I see some notable ITF fighters have a more direct line of their kicking knee. And as any fighter in any style whose been in the business for years would notice, there are a lot of variants to the RH that are subtle, and, no one really uses the crescent much any more (except when we have to do it in a form... lol)
If I were to define the primary differences between the kicks, and therefore their definition, I would describe:
- degree of standing foot rotation
- kicking medium (part of the foot to strike)
- chamber
- direction of the kicking foot
