For an $n$-dimensional extension $K$ of $\mathbb{Q}_p$, we have $K$'s "ring of integers" $\mathcal O_K$ and its uniformizer $\varpi$. We also have the ring of $p$-adic integers $\mathbb{Z}_p$, with $\mathcal O_K$ the integral closure of $\mathbb{Z}_p$ in $K$. I believe that from a basis $v_1, ... , v_n$ of $K/\mathbb{Q}_p$, we can multiply each element by a sufficiently high power of $\varpi$ so that the resulting collection $w_1, ... , w_n$ is an integral basis of $\mathcal O_K$ over $\mathbb{Z}_p$.\
With this in mind, I believe we can extend the notion of a discriminant of an algebraic number field to an extension of $\mathbb{Q}_p$, by defining $Disc(K)$ to be the square of the determinant of the matrix whose $i,j$th entry is $\sigma_i(w_j)$, where $\sigma_1, ... , \sigma_n$ are all the $\mathbb{Q}_p$-embeddings of $K$ into an algebraic closure of $\mathbb{Q}_p$. I
Is this definition appropriate? When is it the same as the polynomial discriminant, and why? For example, when $K$ is an Eisenstein extension of $\mathbb{Q}_p$, with $\varpi$ the root of an Eisenstein polynomial $f$ with other roots $\varpi', \varpi''$ etc., why is it true that $Det( (\sigma_i(w_j) )^2 = (\varpi - \varpi')^2(\varpi - \varpi'')^2(\varpi' - \varpi'')^2 \cdots$?
Yes, this is the usual definition of the discriminant ideal of any finite (separable) extension of Dedekind domains $A\to B$. When $A$ is principal, the discriminant ideal is generated by the discriminant as you defined.
Note that $\mathcal O_K$ is always monogeneous (generated by one element) over $\mathbb Z_p$. More generally, finite separable extensions of complete DVR with separable residue extension are monogeneous, see e.g. Serre, Local fields, III.6.