offshore rig vs drilling rig

offshore rig and drilling rig both are nouns.

Word	Noun	Adjective	Verb	Adverb
offshore rig	Yes	No	No	No
drilling rig	Yes	No	No	No

As nouns, drilling rig is a hypernym of offshore rig; that is, drilling rig is a word with a broader meaning than offshore rig:

offshore rig: drilling rig consisting of an offshore platform (floating or fixed to the sea bed) from which many oil wells can be bored radially
drilling rig: rig used in drilling for oil or gas

Other hypernyms of offshore rig include drill rig, oil rig, oilrig.

offshore rig (noun)	drilling rig (noun)
drilling rig consisting of an offshore platform (floating or fixed to the sea bed) from which many oil wells can be bored radially	rig used in drilling for oil or gas