Is there a term for a group 'owning' a previous insult?

The process is known as reclaiming or sometimes reappropriating.

See uses such as

• https://www.huffpost.com/entry/how-jillian-michaels-is-reclaiming-fag-and-dyke_n_569d1303e4b0ce4964253749

• https://www.facebook.com/1236939913080962/posts/on-behalf-of-all-jews-im-hereby-reclaiming-kike-as-a-word-to-refer-to-a-jewish-b/2313647942076815/

• https://www.huffpost.com/entry/8-women-on-why-they-are-reclaiming-the-word-fat_b_57eec8d0e4b0972364deb2e4

On the flip side 'Faggot' is still insulting and no attempt has been made to 'own' it.

This is quite incorrect, many attempts have been made to reclaim faggot and especially fag.