A displacement field is produced by the polarization of a dielectric. A displacement current is produced by a time-varying electric field.
The two concepts are completely different. A displacement field does not cause displacement current, and a displacement current is not affected by displacement field.
There is no displacement current in a dielectric with a steady electric field because a displacement current is produced by a time-varying electric field, not a steady one. There is a displacement current in free space produced by a time-varying electric field because if there's a time-varying electric field, then there's a displacement current.
Although the two concepts are completely different, they do both have "displacement" in their name, which as CuriousOne mentioned is quite confusing. Furthermore, a displacement current isn't an actual current, with charges moving around and all that; it just has an associated magnetic field as if it were a real current. Hopefully that clears things up.