No that's not quite correct. A lens focal length is as it states and does not change. A 200mm lens is a 200mm lens whether it's on a DX body or FX body.
But what you are getting at is the 'crop factor' of a DX body which gives (with Nikon's DX line) effectively 1.5 times the focal length. The important word is 'effectively', the focal length doesn't change but the FIELD OF VIEW does.
So a 200mm lens will have the expected field of view of a 200mm lens on an FX body BUT will have the field of view of a 300mm lens on a DX body.
The 18-200mm DX lens uses the same measurement ie: focal length as a FX lens. DX lenses are just designed to cover a smaller (DX) sensor and would show dark corners or a dark circle if used on a FX sensor. The 18-200mm lens still is still governed by the crop factor so has an effective (FX) focal length of 27-300mm when used on a DX body.