This is a great question, but I don't think there is an easy answer.
Saito himself proves on p.156 (Cor. 2) that his results imply Ogg's formula, including the missing
case of 2-adic fields. However, the proofs are quite condensed and the underlying technology very advanced.

**Relative dimension 0.**
Saito's approach is motivated by the classical relation between the different, discriminant and
conductor for number fields or local fields. Say $K/{\mathbb Q}_p$ is finite, and
$$
f: X=\textrm{Spec }{\mathcal O}_K \longrightarrow
\textrm{Spec }{\mathbb Z}_p=S.
$$
Classically, there is the *different* (ideal upstairs)
$$
\delta = \{\,x\in K\>|\>\textrm{Tr}(x{\mathcal O}_K)\subset{\mathbb Z}_p\,\}^{-1}
\>\>\subset {\mathcal O}_K,
$$
the *discriminant* (ideal downstairs)
$$
\Delta = (\det \textrm{Tr} (x_i x_j)_{ij})\subset {\mathbb Z}_p,
\quad\qquad x_1,...,x_n\textrm{ any }{\mathbb Z}_p\textrm{-basis of }{\mathcal O}_K
$$
and the *Artin conductor* ${\mathfrak f}({\mathcal O}_K/{\mathbb Z}_p)$ (also an ideal downstairs).
The conductor-discriminant formula in this case says
$$
\textrm{order }(\Delta) = \textrm{order }\textrm{Norm}_{K/{\mathbb Q}_p}(\delta) = \textrm{order }{\mathfrak f}({\mathcal O}_K/{\mathbb Z}_p),
$$
where order is just the valuation: order$(p^n{\mathbb Z}_p)=n$.

Saito interprets these sheaf-theoretically as follows: $\Delta$ is a homomorphism of invertible
${\mathcal O}_S$-modules
$$
\Delta: (\textrm{det }f_*{\mathcal O}_X)^{\otimes 2} \rightarrow {\mathcal O}_S, \qquad
(x_1\wedge...\wedge x_n)\otimes (y_1\wedge...\wedge y_n) \mapsto \det(\mathrm{Tr}_{X/S}(x_iy_j)),
$$
which is an isomorphism on the generic fibre; the classical discriminant is its order = length of the cokernel
on the special fibre.
Then, $-\textrm{order }{\mathfrak f}({\mathcal O}_K/{\mathbb Z}_p)=\textrm{Art}(X/S)$;
this can be defined for any relative scheme over $S$ in terms of l-adic etale cohomology,
$$
\textrm{Art}(X/S) = \chi_{et}(\textrm{generic fibre}) - \chi_{et}(\textrm{special fibre}) -
(\textrm{Swan conductor}).
$$
Finally, the different happens to be the localised first Chern class
$$
\delta = c_1(\Omega_{X/S}) = c_1({\mathcal O}_X\to \omega_{X/S}).
$$

**Relative dimension 1.**
Now we move to one dimension higher, so that $S$ is the same, but $X$ is now a regular model of, say,
an elliptic curve $E/{\mathbb Q}_p$.
The conductor $\textrm{Art}(X/S)$ is still defined, and it is essentially the conductor of $E$,
except that $\chi_{et}(\textrm{special fibre})$ has an $H^2$-contribution from the irreducible components
of the special fibre. To be precise, as explained in Liu Prop. 1
(or using Bloch Lemma 1.2(i)),
$$
-\textrm{Art}(X/S) = n + f - 1,
$$
where $f$ is the the classical conductor exponent of an elliptic curve, and $n$ is the number of components
of the special fibre of the regular model $X$. So, Ogg's formula in this language reads
$$
-\textrm{Art}(X/S) = \textrm{ord }\Delta_{min},
$$
where $\Delta_{min}$ is the discriminant of the minimal Weierstrass model.
So Ogg's formula is like "conductor=discriminant" formula in the number field setting, and Saito proves it
through "conductor=different=discriminant". To be precise, there are three equalities
$$
-\textrm{Art}(X/S) = -\deg c_1(\Omega^1_{X/S}) = \textrm{ord }\Delta_{Del} = \textrm{ord }\Delta_{min}
$$
The first one, "conductor=different" was done by Bloch here (I have no access to this)
and here in 1987. Then $\Delta_{Del}$ is the Deligne discriminant, defined by
Deligne in a letter to Quillen. Analogously to the
sheaf-theoretic interpretation of the discriminant in the relative dimension 0 case,
Deligne constructs a canonical map
$$
\Delta: \det(Rf_*(\omega_{X/S}^{\otimes 2})) \rightarrow \det(Rf_*(\omega_{X/S}))^{\otimes 13}).
$$
It is again an isomorphism on the generic fibre, and Saito calls the (Deligne) discriminant the order
of this map on the special fibre. The second equality, "different=discriminant", is the main result of
Saito's paper, and it is very technical. And finally, Saito on pp.155-156 proves that for elliptic curves,
$\textrm{ord }\Delta_{Del} = \textrm{ord }\Delta_{min}$ (third equality), using properties of minimal
Weierstrass models (at most one singular point), Neron models and existence of a section for models of
elliptic curves.

**Personal note.**
It would be amazing if someone deciphered Saito's proof and wrote it down in elementary terms.
I don't think such a treatment exists, though there are very nice papers by Liu and by Eriksson
on conductors and discriminants.
They treat genus 2 curves and plane curves, respectively, and they are much more accessible.

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