Then why this demarcation? It might just be a case of wrong nomenclature here but can we just do away with this and know the type of photographers with their style apparent in their portfolios? Wedding photography is a mixture of many different genres of photography which includes photojournalism, portraiture, fashion, and lifestyle. The photographer or the team which is apt in more of these departments with an artistic vision more aligned with your taste(which is the most important part IMO) is better suited for your special day.
Many of my clients come with an understanding that a traditional photographer will shoot the group shots of the guests with the couple on the stage and a 'documentation' of the ceremonies as it supplements a candid photographer which in fact is quite opposite from the reality. Having too many teams create a tussle among the photographers for the best spot and hushed arguments during the ceremonies.. Even while trying a different composition, the other photographers would be in the frame and rarely would listen to step back a little for some time. It is optimal to hire one team to cover the wedding from both the sides of the groom and the bride, this way the shoot is more streamlined and well coordinated and the end album you get has a consistent look throughout. If not, it's better to introduce them to each other and ask them to cooperate. I have developed a practice to confirm it with the client that if they need group photographs at the stage, it's better that one member from our team cover it for them.
Having cleared that, I want toaddress the next biggest myth which revolves around the gear. It is quite prevalent in the industry and among the clients that higher end cameras and lenses are the basis for candid photography. So much so that everyone is busy upgrading to next big announcement by camera companies. What we should look at is upgrading our skills, vision, horizon and aesthetics. Sadly these can't be bought off the shelf and is tougher to achieve with no clear way in sight. Coming back to full circle, it should be the work of a photographer that should determine his capabilities. Terry Richardson, a leading fashion photographer uses a point and shoot film camera, much like we had in the 90's. David Burnett used large format film camera to shoot for Olympics, can you believe it? and here somebody's USP is Canon 5D mark4 within a month of it's release. I upgraded to a full frame camera because I wanted to go with wider frames and there aren't many wide angle lenses for crop sensors but before that my investment was first into the lights. Professional equipment makes a job little easier but it in-itself doesn't ensure professional grade results. So it should be immaterial in front of a portfolio.
I hope it clears some air around these terms and general confusion regarding wedding photography. Feel free to comment about your take on it and do share it with someone who would be getting married in near future and could make an informed choice.