I know a number of UK professional acts who travel long distances to do a small folk club gig, for a very low fee, and boost their earnings by busking locally before coming on to the club. Thus the locals get a flavour of the music available at said club, and the acts plug the gig, and often are followed by people who want to hear more. Some of those become regular attenders. If people choose to drop money in an open guitar case, that is called donation, as no arms are being twisted.

Two services rendered for no extra charge.
I wouldn't call that begging.

Local councils have recognised busking as a legitimate art form, and license pitches for the purpose, so I don't think they call it begging either.

As for the guys on the train, they are not busking, but demanding money with thinly disguised menaces. If seen by British Transport Police, they might conceivably be subject to arrest and prosecution.

Don T.