Why are separable and normal field extensions so called?

I always thought, but this is not a historic answer, that "separable" comes from the fact that the roots are "separate" in the sense that there are no repeated roots.

This is in line what I just found when searching Keith Conrad in his lecture notes writes:

The term 'separable' comes from distinctness of the roots: they are separate in the sense that there are no multiple roots.

For "normal" I would say it is pretty arbitrary except that the name should somehow convey that these are the good/well-behaved/regular extension. Sometimes a normal extension is also call quasi-Galois.