The answer is no.

Consider $\mu=\mu_z \circ \mu_y \circ \mu_x$ the blow up of a smooth surface at three points $x$, $y$, $z$, as follows: $x\in X$ is arbitrary, $y\in E_x:=\mu_x^{-1}(x)$, where $\mu_x$ is the blowup of $X$ centered at $x$, and $z$ is the "satellite" point that appears after blowing up $y$, ie, $z=\tilde E_x\cap E_y$, where $E_y:=\mu_x^{-1}(y)$ is the exceptional of the second blowup and $\tilde E_x$ is the strict (birational) transform of $E_x$.

Let $d(\tilde E_x)=d(\tilde E_y)=0$, $d(E_z)=1$, where again $\tilde E_x$ and $\tilde E_y$ denote the strict transforms of the first and second exceptional divisors (but now on $X'$, ie, after blowing up the third). Then in your notation $I_1=I_2=\mathfrak{m}_x$ is the maximal ideal of $x$.