Penny, I'm not sure you understand the difference in the definitions.

Begging:
a solicitation for money or food (especially in the street by an apparently penniless person)

busk

to play music or sing in a public place so that the people who are there will give money

You equated these two. How about: "To work in a garage fixing cars so that the people who are there will give money." The only diffence here is the occupation and the location. Is this begging? No, this is working.
How about "To work in a public place fixing cars so that the people who are there will give money." I've paid mechanics to fix my dead at the side of the road car, while it was still at the side of the road. Is this begging? Once again, no, this is working, only here the only difference is the occupation.
There is a certain social contract inherent in busking. To play on the street, with an open guitar case (even if seeded with a few bucks) is NOT begging. There is an invitation to contribute if you like the music, and to walk if you don't.
Finally, it's not begging if for no other reason that a person who owns a guitar hardly qualifies as "an apparently penniless person."
So let's not only give the buskers a break, let's kick in some cash.