Continuous and discrete-time raised cosine filter properties in frequency domain
Root Raised Cosine has a bandwidth $B=\frac{1+\alpha}{2T_c}$ which is $(1+\alpha)$ times more than the minimum required bandwidth set by Nyquist no-ISI criterion. To satisfy the Nyquist sampling theorem now, one must sample at a rate of $2B$. In time domain, we can say that we have to have more samples/symbol than 1. In practice, and to facilitate the synchronization blocks down the chain, this sampling rate is selected such that we have an integer number of samples/symbol. The smallest integer greater than 1 is 2. In your eq (6), you are sampling at a rate of 1 sample/symbol which induces aliasing. Hence, the results of eq (5) and eq (6) will not be equivalent. For 2 or more samples/symbol for incoming signal and assuming no analog distortions, eq (5) and eq (6) will become equivalent after downsampling to symbol rate.

The best way to understand is to imagine a figure of RRC in frequency domain with aliases at integer multiples of symbol rate $1/T_c$.