In the strict sense of the words, there is no such role as business designer, because at least, for now, nobody, to my knowledge, has come with an integrated blueprint of a business before building it. Businesses have been collectively and organically" designed" and created in "natural" ways before "business designers" have taken the floor.
For that matter, few, if any, are even able to come today with that integrated business architecture or design of a company, if you like, that is the graphical description of a business.
Moreover, the term "business design" is misleading in at least a few ways.
It may remind too many of software design. Do we really want that?
Then, for a new product or capability development process, it means the phase following the architecture modeling, and as a deliverable, the "concretization" of the architecture. Because it would include technology, it is not really the competence of the business.
And it sounds too much like "Design Thinking", pounding at all doors today, which it isn't.
Hence, the term may raise a few eyebrows with the business and in particular with management, in bemusement.
To me, the term "design" looks more like the result of the desire of the business consultants to distance themselves from the "Enterprise Architecture" discipline, its narrow IT scope and its failures. But this is not a good enough reason to invent terms for the sound of it.
Consultants do adopt to a degree the language of their client to smooth communications. But that does not mean that they can do that for terms that denote new disciplines such as Business and Enterprise Architecture. Or take advantage of the appeal of such terms to rebrand their good old occupations.
Moreover, the term architecture sits well with the business because we all relate to the construction and urban architecture while, at the same time, Enterprise Architecture is well known today to the business even if does not inspire confidence.
And while "Enterprise Architecture" may have IT connotations, the term "architecture" does not. While "architecture" may not have been part of the business vocabulary because it springs from the technical domains, the term "design" is even newer to the business. And it falsely reverberates with the technical, artistic or industrial design.
The enterprise business architect does not normally do business or operating models either since they represent "use cases" of EA rather than EA. A savvy architect may do it on demand though.
A business consultant may talk about and do "business and operating models". The client should understand though that this does not equal business or enterprise architecture but rather EA use cases.
One may even call oneself a business designer for the above purpose, even though the naming does not reflect the reality, since the business and operating models are just coming from business out of a collective effort. They should be properly documented based on EA.
Hence, the term business design is rather misleading today as much as "Enterprise Architecture" is when its scope is limited to IT.
I can only hope that business consultancies adjust their language to reality rather than using it for "epater", i.e. to catch eye.
Moreover, the Business Architecture discipline does not necessarily solve business problems by itself as business consultancy does using various business methods. The two complement each other. Business architecture enable business consultants do their job effectively, thoroughly and consistently. While they are different disciplines, one enabling the other, they have different methods.
Business consultants cannot automatically claim that they are BAs or EAs. The reverse is also true in most cases.
The reporting line for a business oriented EA function should be at least at the same level with, rather than reporting to the '"head of "operations" or the head of "business processes" or "transformation" or even "strategic planning' because it has to cover them all.