S1 Form but no deregistration

I work in France and have received my Carte Vitale. I am living in the Netherlands still, and need to be registered there for health insurance. I was told by the Dutch health insurance agency that I need an S1 form to be registered in the Netherlands for my Dutch health care needs. I just called to get the S1 form, but was told that this would deregister me from the French health insurance. But this is not what I want, I want to have in the Netherlands the Dutch health insurance, but be covered in France and other places using my Carte Vitale.

The Dutch organization told me that with an S1 form I should not be deregistered from France. Have I misunderstood the French operator, and is this indeed the case?