Self cutting hose connector piece of nonsense, the big sheds sell them for DIY outside taps, as you mention.
As boss mentions, undo the clamp and cut out the pierced section.
It may be easier as a DIY fix to just cut out the pierced part and slot a compression T in there, it should be just about bang on size wise. Then add in a tail of copper off the Tee and to do it right a single check valve and then the W/M valve which has a 3/4" end that the white connector will fit onto.