**Edit:** Following Will Sawin's comments, I realized that my first answer contained many mistakes. I try to emend my answer, writing $G_\mathbb{Q}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $G_K=\operatorname{Gal}(\overline{\mathbb{Q}}/K)$ throughout.

First of all, your $r$ is actually equal to $1$: indeed, if you look at Theorem II.9.1 in Silverman's *Advanced Topics in the Arithmetic of Elliptic Curves*, you see that what he calls $\alpha_{E/L}$ is a character which is trivial when restricted to infinite idèles (use property (i) there), and therefore the property in Theorem II.9.2 defining the Grössencharakter, namely $\psi_{E/L}(x)=\alpha_{E/L}(x)N^L_K(x^{-1})_\infty$, shows that $\psi_{E/L}$ is of infinite type $(-1,0)$. Secondly, I think that you have a sign-problem: you should have the same sign in the infinity-type (negative) and in the tensor power showing up in your equation.

Secondly, you say in the comments that you took the above equation from Kato's 2004 *Astérisque* paper *$p$-adic Hodge theory and values of zeta functions of modular forms* but beware that in his setting $K$ is imaginary quadratic (see the beginning of §15) and that he is working with a classical modular form (of some weight $k$): it is certainly a good idea to try to grasp what he's doing in case $k=2$, but this corresponds to an elliptic curve $E$ precisely when the curve is defined over $\mathbb{Q}$, so that $K$ is an imaginary quadratic field of class number $1$ (assuming that $E$ has complex multiplication by the full ring of integers). A final remark about Kato's paper: although on page 160 he says that $``$as representation of $G_\mathbb{Q}$ there is an isomorphism
$$
V_\ell(E)\cong V_{\ell}(\psi)\oplus \sigma V_\ell(\psi)"
$$
he **does not mean** that the above is a direct sum of representations. Indeed, in his notation three lines below his statement, if you pick $(x,0)\in V_\ell(E)$ and $\gamma=\iota\tau\notin G_K$, you get $\gamma(x,0)=(0,\gamma x)$, showing that the direct sum is not a sum of representation. Indeed, as Will Sawin observed, the Galois representation $V_\ell(E)$ is *irreducible*.

I consider then an elliptic curve $E/\mathbb{Q}$ with CM by $\mathcal{O}_K$. I write $E_{/K}$ for its base-change to $K$ and I write $V_\ell(E),V_\ell(E_{/K})$ for the respective Tate modules. They are somehow different creatures: the first is endowed with an action of $G_\mathbb{Q}$ while the second has an action of the subgroup $G_K$ and is, moreover, a free $\mathbb{Q}_\ell\otimes K=K_\ell$-module of rank $1$, as discussed for instance in Tate module of CM elliptic curves: this is not true for $V_\ell(E)$, because the complex multiplication is not defined over $\mathbb{Q}$ and, in particular, it does not commute with the Galois action of the full group $G_\mathbb{Q}$. Let's analyze these two representations separately:

As said, $V_\ell(E_{/K})$ is of rank $1$ over $K_\ell$ and I claim that it is isomorphic to $K_\ell(\psi)$, which is the rank-$1$-module $K_\ell$ endowed with an action of $G_K$ given by $g\cdot v=\psi(g)v$, seeing $\psi(g)\in K^\times\subseteq K_\ell^\times$. This is the commutativity of the diagram in Theorem 9.1 (ii) of Silverman's *Advanced topics...*, together with the definition of $\psi$ in Theorem 9.2 *ibid.*. As a by-product, you get an isomorphism
$$
K_\ell(\psi)\cong H^1(E(\mathbb{C}),\mathbb{Q}_\ell)^{-1}
$$
because on the one hand
$$
H^1(E(\mathbb{C}),\mathbb{Q}_\ell)^{-1}\cong H^1_\mathrm{et}(E_{/K},\mathbb{Q}_\ell)^{-1}\cong V_\ell(E_{/K})
$$
(this is well-explained here Etale cohomology and l-adic Tate modules, upon realizing that $(\ast\otimes -1)$ means ``taking the dual''); and, on the other, because
$$
H^1(E(\mathbb{C}),\mathbb{Q}_\ell)^{-1}=\mathbb{Q}_\ell\otimes_{\mathbb{Q}}H^1(E(\mathbb{C}),\mathbb{Q})^{-1}=\mathbb{Q}_\ell\otimes_{\mathbb{Z}}H_1(E(\mathbb{C}),\mathbb{Z})=\mathbb{Q}_\ell\otimes \mathcal{O}_K
$$
where the last isomorphism is the map $\gamma\mapsto \int{_\gamma} {\omega_E}$ (in general, the image of this map is a lattice $\Lambda$ such that $E(\mathbb{C})=\mathbb{C}/\Lambda$, so in this case $\Lambda=\mathcal{O}_K$).

The tensor product $V_\ell(E)\otimes_{\mathbb{Q}_\ell}K_\ell$ has rank $2$ over $K_\ell$ and $G_\mathbb{Q}$ acts on it on the first factor. It turns out that there is an isomorphism of $G_\mathbb{Q}$-representations
$$
V_\ell(E)\otimes_{\mathbb{Q}_\ell}K_\ell\cong \operatorname{Ind}^K_\mathbb{Q}\big(K_\ell(\psi)\big)
$$
as proven (in the general setting of CM cuspforms) in Ribet's 1977 paper *Galois representations attached to eigenforms with Nebentypus*, Theorem 4.5. You can retrace the line of Ribet's argument (who was inspired by the classical proof for elliptic curves) by doing exercices 2.29-2.32 in Silverman's *Advanced topics...*: the main point is to compare the traces of both representations and then to invoke a general result saying that two $2$-dimensional irreducible representations sharing the same trace are isomorphic.

Now, by restricting the $G_\mathbb{Q}$-representation constructed above to the subgroup $G_K$ you **don't find** $V_\ell(E_{/K})$ (which is of rank $1$ over $K_\ell$), but you rather find
$$
\operatorname{Res}_{G_K}\big(V_\ell(E)\otimes K_\ell\big)\cong\operatorname{Res}_{G_K}\operatorname{Ind}^K_\mathbb{Q}\big(K_\ell(\psi)\big)
$$
and you can apply Proposition 22 of Serre's *Représentations linéaires des groupes finis* to deduce that this latter object is $K_\ell(\psi)\oplus \sigma K_\ell(\psi)$, in turn isomorphic to $$
\Big(H^1(E(\mathbb{C}),\mathbb{Q}_\ell)^{-1}\Big)\oplus \sigma \Big(H^1(E(\mathbb{C}),\mathbb{Q}_\ell)^{-1}\Big)
$$
by point 1. above.

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