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.
