equivalent characterizations of discrete valuation rings

I think you will forgive me that I will not have a close look on all your arguments, but I can focus on the last statement from wikiepdia.

One direction is clear: If $(R, \pi)$ is a DVR, every fractional ideal is of the form $R\pi^{n}$ with $n \in \mathbb Z$, thus they are linearly ordered and in particular the intersection of two of those will always be the smaller one.

For the other direction, the property clearly implies that $R$ is local, because the intersection of two maximal ideals is a proper subset of both maximal ideals. More generally, the set of all fractional ideals is linearly ordered, because $I \cap J \in \{I,J\}$. In particular the set of all sub-vectorspaces of $\mathfrak m/\mathfrak m^2$ is linearly ordered, this of course enforces it to be one-dimensional. Hence $\mathfrak m$ is principal.

Summarizing, a DVR is a noetherian domain which is not a field and whose set of fractional ideals is linearly ordered. The statement on wikipedia with the intersections is just a reformulation.