I think that the point lies in the difference between a primitive and imprimitive $L$-function. Before entering the details, let me observe that Washington's definition of $p$-adic $L$-functions (as the one found in Washington's Introduction to Cyclotomic fields, Theorem 5.11) gives the interpolation
$$
L_p(1-n,\chi)=(1-\chi\omega^{-n}(p)p^{n-1})L(1-n,\chi)\qquad n\geq 1
$$
and this is shown to be equivalent to the Kummer congruences that you quote, namely
$$
(1-\chi\omega^{-n}(p)p^{n-1})\frac{B_{\chi\omega^{-n},n}}{n}\equiv (1-\chi\omega^{-m}(p)p^{m-1})\frac{B_{\chi\omega^{-m},m}}{m}\pmod{p^a}
$$
whenever $m\equiv n\pmod{p^{a-1}}$ and $\chi=\omega^r$ is a power of the Teichmüller character. The value $\chi\omega^{-n}(p)=\omega^{r-n}(p)$ is considered, in both statements, as the value of the primitive character attached to $\omega^{r-n}$ at $p$, in the following sense (the discussion is taken from Chapter 3 of Washington's book). When $r\not\equiv n\pmod{p-1}$, the character $\omega^{r-n}$ has conductor $p$ and is primitive; the value $\omega^{r-n}(p)$ is $0$, hence the "correction factor" disappears. The interesting part arises when $r\equiv n\pmod{p-1}$, so that $\omega^{r-n}$ is the trivial character of conductor $p$: this is an imprimitive character, whose primitivisation is the trivial character of conductor $1$, whose value at $p$ is $1$: the "correction factor" is there and takes the value $(1-p^{n-1})$. This, as Washington remarks right after his Theorem 5.11 is the Euler factor at $p$ of the complex $L$-function defined on page 31 of his book ($2^\text{nd}$ edition), where the product runs over all primes.
In Deligne--Ribet the situation is different, and so is in Ribet's report that you quote. Indeed, in equation $(1.1)$ of his report he defines the $L$-function of a character $\varepsilon$ on a ray class group $G_\mathfrak{f}$ as a Dirichlet series over all prime-to-$\mathfrak{f}$ integral ideals. In section 4 of the report, he applies this to the limit $G_{\mathfrak{f}\mathfrak{p}^\infty}$. To make the bridge with Washington's situation, take $K=\mathbb{Q}$ as base field, and $\mathfrak{f}=1$. As character, chose $\varepsilon=\omega^{r-n}$ with $r\equiv n\pmod{p-1}$ (since otherwise, as discussed, Washington's and Ribet's expressions coincide). Then the $L$-function considered by Ribet is given by
$$
L(s,\omega^{r-n})=\prod_{\ell\neq p}(1-\omega^{r-n}(\ell)\ell^{-s})^{-1}\qquad\operatorname{Re}(s)>1
$$
because $\omega^{r-n}$ is seen as character on $G=G_{p^\infty}$ of conductor $p$, and the sum defining $L(s,\omega^{r-n})$ is performed only over primes $\ell\neq p$. The corresponding Kummer congruences get modified accordingly.
It might be helpful, to convince yourself of this discussion, to compare the interpolation formula proposed by Washington in Theorem 5.11 and the one proposed by Ribet in his final section on page 13, in the paragraph preceeding equation $(4.6)$: he claims that $L_p(1-k,\varepsilon)=L(1-k,\varepsilon\omega^{-k})$, showing the difference with Washington's.
