PennyBlack:
I have several problems with your definitions. First, there is the problem that you see them as being equivalents. I see them as being definitions by genus and species, in the Aristotelian method.   Your definitions were:

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

The genus in begging is a solicitation for money or food. The species (i.e. that which differentiates begging from any other solicitation of money or food) is "by an apparently penniless person.

The genus in busking is to play music or sing in a public place. The species is that the people who are there will give them money.

From a strict logician's point of view, even using your definitions, these are not equivalents.

Secondly, the real problem is that you got your definition of busk from the Cambridge Advanced Learner's Dictionary. A quick search of the Internet tells us that:

"The new Cambridge Advanced Learner's Dictionary gives you everything you need to learn English and pass exams!
It's bang up to date, easy to use and comes with a great CD-ROM which takes your English to another level."

In other words, it's a student dictionary. Let's look at some other, perhaps more in depth sources.

The Concise Oxford English Dictionary (www.askoxford.com) defines Busk as:

busk: verb 1. play music in the street for voluntary donations.

voluntary: adjective 1 done, given, or acting of one's own free will.

Whereas beg is defined:

Beg:   • verb (begged, begging) 1 ask earnestly or humbly for something. 2 ask for food or money as charity.

Webster's 1913 Dictionary defines busker as:

n. 1. a person who entertains people for money in public places (as by singing or dancing)

WordNet Dictionary defines busker as:

Noun- a person who entertains people for money in public places (as by singing or dancing.)

So, if you want to keep begging when you're dancing, go right ahead. We'll try not to let you know you're busking.