About I/Q signals in Rx and Demodulator

Fig 1. Hardware diagram of and I/Q Modulator and Mathematical Background

In low-IF receiver, the quadrature LO signals are generally used in order to dissect RF signals into baseband I/Q signals and further used for image rejection.
From what I've known, in homodyne receiver when we dissect RF signals into I/Q with quadrature LO mixing, the I/Q signals are totally distinguished and each represents the I and Q information.

Today, one of my classmates told me that the I/Q signals in receiver are different from those of demodulator.
What she said is that the signals on the baseband I/Q path in the receiver are not seperated into the I/Q data shown in Fig 1. Her opinion is that the I/Q in receiver are a little different from the I/Q data in Fig 1 and the signals on each I/Q path in Rx are still the combination of IQ data(Icos(2*pi*fc*t)-Qsin(2*pi*fc*t)) as shown in Fig 1 and finally the I/Q data seperation will be done in demodulator.

Is her opinion right?

I am getting confused because if she is wrong, what is the role of demodulator after homodyne receiver using quadrature mixing thus having I/Q baseband path?