"
CL is a function of angle of attack." - yes. And as long as we're not near the stall, and it's a high aspect ratio, low sweep wing, a pretty simple function at that. Say, it looks something like this. This wing stalls (say, the red Reynolds) at 11 degrees AOA and 1.2 CL. Below 9 degrees, it's a linear relationship of 0.1 CL per degree AOA. Multiply or divide one by 10, you get the other. This is what I meant by saying that from the pilot's perspective, they're essentially synonymous (except near the stall). Just as you can think of the elevator as the AOA control, you can think of the elevator as the CL control. Pull back and you increase the CL - that is, until shortly before the stall. (Then, it flattens out, and after that, starts decreasing).
"You don't see CL plotted vs. speed - it's vs. AOA." Also true.
But,
"It's not a function of speed." - not necessarily true. If we constrain to a 1G (or any constant G) condition, AOA becomes a function of speed, so therefore, CL is then also a function of speed.
"Yes, in level flight, lift stays constant - pretty much by definition. But if you speed up, the CL doesn't change, the AOA does - which moves the CL to a different point on the CL/AOA curve."
First you say the CL doesn't change, (false) but then you say it moves to a different point - which is true, and which means it changed. Let's say on the above graph we start stable at some mid condition, at 5 degrees AOA, and 0.7 CL. And, filling out some constants, this means 100 knots. If we move the stick slightly forward, let everything stabilize, the AOA will decrease - say by one degree so now we're at 4 degrees AOA, and the CL has changed to 0.6. And, instead of 100 knots, now this yields an increase to 110 knots.
"CL is a dimensionless term that measures the ability of a particular body shape to generate lift." What I'm getting is that you're using CL to mean
maximum CL, or CLmax. CLmax, yes, shows the "ability" to make lift. The liftier the wing (high camber, high thickness, etc.) the higher the CLmax is. But the CL can be (and, most of the time, is) at any value below the maximum, as dictated by the current conditions (and commanded by the stab and elevator). I.e., CL represents the
current lift, not the ability of maximum lift (that would be CLmax). If you take the lift equation and cross out all the constants (for straightline flight, in a given situation), it's plain that CL varies inversely with speed.
L = CL *
0.5 *
density * speed^2 *
area