Is this "certificate of sponsorship" different to a 'work permit'?

If so, did these musicians have their work permits? Every tour has to have a 'sponsor' of some kind. Usually it's a booking agent who fulfills this role and they know how to process these things. There's no such thing as a private or social visit if remuneration is being made for musical performances.

I know of quite a few cases of musicians deported because they didn't have work permits. One renowned American singer-songwriter was caught out quite recently arriving for gigs without a permit.

When questioned if you're doing gigs it's best to admit it. Even customs officers have enough nous about them to go into another room, switch on a computer, google your name, find your website and know that your telling porkies.