The way I think about business coaching is perhaps the more traditional way of coaching. On an individual basis it is the objective improvement of someone in some business or profession as measured by business results and behaviors. Business coaching focuses on helping the individual grow as a person AND learn and improve in the context of the business, trade or profession in which they earn their living.
While there are a number of coaching programs with exciting descriptive adjective, like Optimal, Exponential, Transformational, Quantum-Leadership etc. - they are business, trade or profession agnostic.
As an entrepreneur and corporate executive I have found that truly effective coaching in a business or field requires deep knowledge and expertise in that business. This is still common practice in many fields like medicine/health-care, aviation and most of the high skilled trades and professions like accounting and law.
It is not merely historic convention that a professional football team hires a coach with football knowledge and expertise - it is because they truly understand, can relate and most importantly can help them produce the best results for the teams and players.