What is meant by the "cosine corrected response" of a pyranometer? I have done an experiment that involved placing a pyranometer on a flat area and measuring the solar irradiance. The pyranometer used was a Kipp & Zonen CMP3

The irradiance1 is proportional to $\cos(Z)$, the cosine of the solar zenith angle $Z$ (SZA). The SZA is the angle between the vertical and the sun.
Is the output of the pyranometer proportional to $\cos(Z)$? As far as I can tell, having read the manual for this pyranometer, and looking on the internet, yes. The manual says

Ideally a pyranometer has a directional response which is exactly the same as the cosine-law

However, the marker of my report says that as glass dome receives light from 2 pi steradians, the output is independent of $\cos(Z)$. This seems completely wrong to me.
This is what he said:

The pyranometer has a $2\pi$ field of view due to its hemispherical glass dome.
Consider a parallel beam incident on a normal surface, with intensity $F_0$. When the beam is not normal to the surface, the intensity perpendicular to the surface is now reduced and is $F_0 \cos(Z)$. The cosine corrected response of the pyranometer means that the output of the pyranometer is $F_0$ for the case of normal incidence and for the case where the angle is $Z$. This is why the pyranometer has a curved surface.
As the pyranometer manual states, this correction is valid up until $Z$ = 80 deg. You have misinterpreted what the manual refers to as a ‘cosine corrected response’.

As far as I can tell, the glass dome is simply there to protect the flat thermopile sensor, and is hemispherical so that sun rays are perpendicular to its surface and no refraction occurs. The output is surely due to the sun rays striking the thermopile, and therefore depends on the SZA?
And what would be the point of a pyranometer whose output is independent of the SZA? Surely the irradiance at the surface is the important quantity one would normally try to measure (the lower the sun is in the sky, the smaller the irradiance)?
1Defined as the radiant flux (power) received by a surface per unit area, measured in watts per square metre (W/m^2) by Wikipedia
 A: This is from a brochure on pyranometers found on Kipp & Zonen's infomation page about the CMP3:

A pyranometer measures the global horizontal solar irradiance (GHI) which is composed of diffuse horizontal solar irradiance (DHI) from the
  sky and direct normal solar irradiance (DNI) from the sun. If shaded
  from the direct sun a pyranometer measures diffuse horizontal solar
  irradiance (DHI).

The global horizontal solar irradiance is defined (link) as the energy received by a horizontal surface, which has a $cos(Z)$ dependence. So you are right, the pyranometer does have a $cos(Z)$ dependence.
Update: I just found another brochure on the same site which states this explicitly:

GHI = DHI + DNI • cos(θ), where θ is the solar zenith angle (vertically above the location is 0 °, horizontal is 90 °). GHI is measured by a pyranometer mounted horizontally.

You also asked why someone might want an instrument without a $cos(Z)$ dependence. Well, not everyone is interested in absorption by flat surfaces with a well-defined orientation. For example, absorption by a collection of randomly oriented surfaces (e.g. leaves on a tree) will not have a $cos(Z)$ dependence.
