If one is working on a non-B I am led to believe that he will need to leave the country when quitting work - even though the visa is still current. I am also led to believe that if one doesn't have a work permit, then one can stay in the country until the visa expires (after quitting work) because there will be no official record of that person leaving the job.

Can anybody shed any light on this matter please?

I you have a an entry obtained by a visa you do not have to leave. If the visa is still valid will it will also remain valid.

If you have an extension of stay obtained at immigration you must leave when your job ends.