china pink vs garden pink

china pink and garden pink both are nouns.

Word	Noun	Adjective	Verb	Adverb
china pink	Yes	No	No	No
garden pink	Yes	No	No	No

As nouns, garden pink is a hypernym of china pink; that is, garden pink is a word with a broader meaning than china pink:

china pink: Chinese pink with deeply toothed rose-lilac flowers with a purplish eye; usually raised as an annual
garden pink: any of various flowers of plants of the genus Dianthus cultivated for their fragrant flowers

Other hypernyms of china pink include pink.

china pink (noun)	garden pink (noun)
Chinese pink with deeply toothed rose-lilac flowers with a purplish eye; usually raised as an annual	any of various flowers of plants of the genus Dianthus cultivated for their fragrant flowers