Caporegime
I think it does make a fundamental difference, he may have just being saying 'put Britain first' as a statement about the importance of British sovereignty and nothing to do with the group called 'Britain First' or their rhetoric.
By their rhetoric, I mean the same set of ideas they espouse not necessarily the words they've said. The same rhetoric that certain posters on this forum have used in describing Remain voters as "traitors", the same rhetoric that Farage espouses when he describes how he wouldn't want a Romanian for a neighbour or presents posters of a line of Syrian refugees with "bursting point" written above it.