A "bank nurse" is used in the British National Health Service NHS to refer to people who are used to work at short notice to work in hospitals for when they are needed. So it would be the same as being on call, but a team is on call in a "nurse bank" not a single employee; that is employed directly by the hospital. A bank can be used for anything. Common ones are "sperm banks, animal and human" "seed banks" "blood banks" "industrial banks" "agricultural banks" "investment banks"
employment agencies that supply nurses only, is in effective a "nurse bank".
"staff nurse" = A nurse of senior rank just below a Sister, Sister I believe may have come from the origins of the word "hospitable" when they were run by medieval nuns, the sister is the boss. "hospitable, hospital, hospice, directly related words that were the same in meaning.
"registered nurse" = a nurse that is registered with a professional or government council. Has passed the minimum qualifications. There would be other organisations to register with and qualifications to pass. Countries and states would have their own organisations and qualifications. The UK registration system is a little different and older beginning with "florence Nightingale" who began opening schools, and training programs after returning from the Crimean War. But the idea is the same.