For me personally, the phrase "anti-gay group" muddies the waters because although Salvation Army (1) does (or did) have some anti-gay policies, and (2) is a "g

For me personally, the phrase "anti-gay group" muddies the waters because although Salvation Army (1) does (or did) have some anti-gay policies, and (2) is a "group", its main purpose is to be a food or shelter charity; not to pursue the disenfranchisement of gays through say, political compaigning. To illustrate, I might refer to an organization under religion X as an "anti-women religious group" because perhaps religion X had some policy on women's activites that I believed were sexist; then for me "anti-women religious group" would be a technically correct phrase in that X is "anti-women" and "a religious group", but the phrasing would probably be objectionable to practitioners of the religion who don't see those particular policies as fundamental to their religion, and perhaps didn't even consider the policies sexist in the first place. In general, I don't think I often see this loose use of terminology applied to other cases; like I don't think people call Nike a pro-child-labor company, although if that were the topic of discussion I might say Nike uses child labor. To me it's about avoiding ambiguity.