Sophie, you're tending to overthink this a little, methinks.
First, a useful little mnemonic - which appears to date back to the quite early days of US military aviation - "iced tea is a pretty cool drink" to cover the order of things -
ICEd Tea is a Pretty Cool Drink -
Indicated - Calibrated - Equivalent - True
is a Position - Compressibility - Density
gives the sequence IAS- position error - CAS - compressibility correction - EAS - density correction - TAS
IAS - what the box reads
position error - covers things like mechanical instrument errors, pressure lag errors in the plumbing, static source position not picking up a consistently accurate static pressure from outside, etc
CAS - in effect sort of a "true" value of IAS
compressibility correction - as the Mach Number gets above trivial values (say, 0.2 - 0.3) the air flow to the pitot starts to see the air "squashed up" a bit and the ASI starts to over read so that the value we use (CAS) progressively gets into an erroneous value. There is a standard way of correcting for this to arrive at EAS. For low performance aircraft (ie low altitude and low speed) Mach Number is pretty low and, so, we ignore compressibility corrections and EAS for typical training aircraft.
EAS - this is the speed on which we base load calculations as it is the best value we have to represent the real airflow around the aircraft. For low performance aircraft, such as the typical training aircraft, we can ignore it altogether and consider that CAS = EAS
density correction - the ASI will read more or less correctly at standard (ISA) sea level conditions. As we move away from ISA sea level conditions, the errors start to mount. Fortunately, the correction is very simple. It is useful for pilots to know the material as this gives us a way to figure out local atmospheric density, should you ever have a need to know such stuff.
The equation to correct EAS to TAS is shown below. As we are concerned with low performance aircraft, we will ignore EAS and use CAS in its place.
TAS = CAS/√σ
where sigma (σ) = density ratio = ρ/ρo
where ρ (rho) is the local density and ρo is the standard sea level density (1.225 kg/m3).
As an aside, when we play with performance stuff, we make considerable use of non-dimensionalised numbers (ratios, in this case). You may come across references to sigma (as above), delta (δ - ratio of pressures), and theta (Θ - ratio of temperatures) in your general aviation reading.
Now if we know CAS and sigma, for our low performance aircraft flying along at whatever height, then we can figure out the conversion from CAS to TAS. For low speed (ie non-compressible) flight, any of the navigation computers will do this for us.
Bosi, (that nice chap) has posted a scan of a Dalton (E6B) computer with this stuff shown, so we will use his scan for an example.
Now, if we were to look up a table of ISA values (you can get these from the net via Mr Google), we would find that the density at 10,000 ft is 0.7385 kg/m3. Dividing this by standard sea level density would give us a sigma value and, then, √σ = 0.8594. If we take the reciprocal of this, we get 1/√σ = 1.1637.
Looking at Bosi's scan, the airspeed cutout is set to give a density height of 10,000 ft. This setting (known as a nomogram), while providing the density height for us, is not intended solely for that purpose. Its main purpose is to set the outer slide rule C/D scales to specific values.
If you look at the settings Bosi has ringed, at the position where the outside scale is "10" shows, on the inner scale, a value of 0.86. Notice that this is the value we worked out, above, for √σ = 0.8594, rounded off to two decimals for the slide rule precision.
Similarly, if you look at the position where the inner scale is "10", this shows, on the outer scale, 1.161. Again, notice that this is the value we worked out, above, for 1/√σ = 1.1637. The discrepancy simply reflects the fact that we can't read the third decimal all that accurately on the slide rule.
So, what the airspeed cutout is doing, is setting up the multiplications
TAS = CAS/√σ , and
CAS = TAS x √σ
depending on which way you want to do things - same result, though.
So, we can use the whizz wheel to get the value of sigma for any height and then multiple this by the standard sea level pressure to get the density at that height. You can use this to impress your colleagues at the Aero Club bar on a Saturday afternoon, should you so desire ....
Just a couple of clarifications, though, if I may -
1. IAS (indicated) is what is measured at the pitot tube, how fast is wind blowing through the pitot tube.
Not quite. The pitot system is sealed so that there is no flow through it. What it is doing is bringing the incoming airflow to a standstill which results in an increase in pressure. The pitot gives us static + dynamic pressure. By using the static input from the static port, we can get rid of the static pressure component and end up with dynamic pressure, from which we can figure out speed.
2. .... For light aircrafts the difference is minor, and likely you won't find it in e.g. a Cessna or Piper, therefore CAS = IAS.
Again, not quite. The driver in calibration is the altimeter, for which there are error limits. Once these are met, the end result is that the ASI errors will be small and, sometimes, zero. All aircraft will have a PEC card, including Cessnas and Pipers. Because the PEC is small for the ASI (except around the stall), we usually can approximate CAS to IAS without too much problem.
3. ..... corrected for air pressure/density/temperature.
That is to say, corrected for pressure/temperature, or density.
) 
