Hi Slim!
If, hypothetically, I were in the mood to have a discussion on "purpose" of a bicycle, I would further narrow your three classes to two: functional and recreational. And all recreational bikes have functional aspects, and all functional bikes have recreational aspects. And all of these aspects are somewhat objective, but largely subjective. My functional commuting bike functions for me with the inclusion of a backpack, but might fall into someone's definitional of a recreational bike.
I have, and I must apologize for this, read many of your threads. And many of them are very humorous, entertaining, witty, and ultimately very "West Coast", but they largely rely on pigeon holing a rider or a bike or a material or a process into that specific nook that you believe is the correct fit. You are very black and white in your summaries, and I believe firm in your beliefs and commitments. Just recognize that many of us who embrace the "anything you ride is good as long as you ride" mentality will gag and vomit on your pablum.
Luckily, this was all a hypothetical discussion, and I wasn't in the mood to share.