I study a spectral efficiency of a system with a precoding scheme with different channel schemes.
Algorithm A for channel estimation gives a smaller estimation error instead of Algorithm B.
I have computed the precoding schemes using the resulted estimated from Alg. A and Alg.B. As a result, the capacity with precoding for Algorithm A is better ( bigger) than for Algorithm B.
The channel capacity expression you cited is the one with CSI at receiver that means the receiver know perfectly the realization of fading channel, denoted $\mathbf{H}$, but not the realization of additive white Gaussian noise. In that case, the channel capacity is the max (or sup) data rate at which there exists a receiver that achieves decoding error probability smaller than any infinitesimal $\epsilon > 0$, and can be computed using the expression in your links (with adding expected value operator depending assumptions) see chapter 8.
Therefore, I would not be convinced if people say their receiver needs to estimate the channel while it is assumed to be known (that estimation is not illegal though).
That said, it is legit if the channel estimation is not at the receiver. One such case is that the channel estimation is done at transmitter. But in any case, receiver must known perfectly the channel so that you can use the channel capacity expression.
Thus, if your transmitter estimates the channel and calculate precoding vectors to transmit data and your receiver knows perfectly the precoding and the channel, you can use the capacity expression.
