discrete signals - Link between DFS, DFT, DTFT

My understanding of DFT is as follows

For a signal $x[n]$ of finite-length, the DFT is DFS of the periodic extension, $\tilde{x}[n]$, of that signal $x[n]$ and also another way to view DFT is that it’s a sampling of continuous DTFT.

Given that it is possible to reconstruct a original signal from sampled signal, provided the sampling is greater than Nyquist frequency. We know that the DTFT for sampled signal is a series of replications of the spectrum of the original signal at frequencies spaced by the sampling frequency. Now, since DTFT is continuous and periodic, we can further breakdown DTFT at intervals and still be possible to reconstruct the DTFT and consequently the original signal. This act of breaking down or sampling the DTFT is called DFT.

Is my interpretation of DFT correct? I would welcome any (true) facts or implications to test my understanding