I can not concede that the underpinning meaning of completing a business registration means as you propose is - "my name is ____ ____, and I am currently operating this account as the business ______", 

lol - maybe for a small sole trader business DG, but not for a larger business partnership or company - just not appropriate, by any measure, and certainly not legally, as it would seek to negate the legal protections that are afforded to an individual under a number of business structures.