What's the difference between "Discrete Fourier Series" and "Discrete Fourier Transform"?
They are the same thing. Just the same. 

But for some reasons, we want to call the first period of the DFS sequences $\tilde{x}[n]$ and $\tilde{X}[k]$, as the new DFT sequences $x[n]$, $X[k]$.

They are both periodic. The periodicity of DFS is natural extension of periodicity of complex exponential and periodicity of time sequence, but the periodicity of DFT is also forced by the modulus operator. 

Be careful! DFT is also linked with finite length aperiodic sequences $x[n]$ and their DTFT via the frequency sampling relation; i.e., $X[k] = X(e^{j \frac{2\pi}{N}k})$.

Because of this latter relation of the DFT $X[k]$ to the DTFT $X(e^{j \omega})$ of aperiodic (and finite length) sequences $x[n]$, in order to remember and imply the inherent periodicity in DFT (due to its definition through DFS) we shall enforce the modulus operator into the arguments of DFT sequences.