Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion points of $E$. Then if $m=p_1^{r_1}\cdots p_k^{r_k}$ where $p_i$, $i=1,2,\cdots,k$ are prime numbers, can we say that $Gal(K(E[m])/K)\cong Gal(K(E[p_1^{r_1}])/K)\times\cdots\times Gal(K(E[p_k^{r_k}])/K)$?

If yes, then is it easier to directly prove the isomorphism above or the isomorphism $Gal(K(E[p_1^{r_1}])\cdots K(E[p_k^{r_k}])/K)\cong Gal(K(E[p_1^{r_1}])/K)\times\cdots\times Gal(K(E[p_k^{r_k}])/K)$?

I understand that the second isomorphism will only hold if $K(E[p_i^{r_i}])\cap K(E[p_j^{r_j}])=K$, where $1\leq i,j\leq k$, $i\neq j$ and in fact this actually is the case for some non-CM elliptic curves which I recently found in the book "The decomposition of primes torsion point fields" by Clemens Adelmann. But does it also hold for curves in the CM case?

My intuition:

Since it is true that $K(E[m])\cong K(E[p_1^{r_1}])\cdots K(E[p_k^{r_k}])$, i.e, it is isomorphic to the compositum of the field extensions $K(E[p_i^{r_i}])$, $1\leq i\leq k$, so these extensions $K(E[p_i^{r_i}])$ are linearly disjoint from each other and that would also imply that the intersection of any two of these $(i\neq j)$ is $K$?

Is this correct?