nor does the President "interpret" the law; that is the domain of the Judiciary.
That is incorrect, the president’s responsibility is to execute the laws of the United States. In order to execute a law, one must interpret it.
The exclusive domain of the judiciary is the power to adjudicate cases. In order to adjudicate a case involving a law, one must also interpret.
> In order to execute a law, one must interpret it.
I mean, yes, but in a non-judicial sense. The judiciary has the constitutionally assigned duty to give binding interpretations in the course of adjudicating cases.
> In order to execute a law, one must interpret it.
This isn't true.
> The exclusive domain of the judiciary is the power to adjudicate cases. In order to adjudicate a case involving a law, one must also interpret.
This is actually true.