public press vs mag

public press and mag both are nouns.

Word	Noun	Adjective	Verb	Adverb
public press	Yes	No	No	No
mag	Yes	No	No	No

As nouns, mag is a hyponym of public press; that is, mag is a word with a more specific, narrower meaning than public press:

public press: the print media responsible for gathering and publishing news in the form of newspapers or magazines
mag: a periodic publication containing pictures and stories and articles of interest to those who purchase it or subscribe to it

Other hyponyms of public press include free press, newspaper, paper, magazine.

public press (noun)	mag (noun)
the print media responsible for gathering and publishing news in the form of newspapers or magazines	a periodic publication containing pictures and stories and articles of interest to those who purchase it or subscribe to it