My definition of a smartphone is a phone with PDA functions (calendar, contacts, email, internet). The first smartphones were basically PDA's with a cell phone function tacked on, such as the first Palm phones. By that definition, the iPhone is certainly a smartphone, one of the most powerful around IMO. I can't see how anyone could say it is not, though I do see the same perception issue that many others are noticing.
I have used a smartphone for work + personal needs for years. I'm a smartphone power user. When I replaced my Motorola Q smartphone with an iPhone 3G a few months ago, several coworkers looked at me funny like "why are you using a toy like that for business?" Lots of people seem to thing it's a toy. I guess they think that because it's a decedent of the iPod and because casual users and even some kids choose the iPhone (those same people would not choose a Blackberry). The funny thing is, I do have several coworkers who have iPhones but they don't really use them for business, just for fun. I'm probably the only one here who uses it seriously.
I also have another coworker who is getting a Blackberry because "that's what our clients use and it just seems more professional". Well, yes I suppose it is more professional, but also more convoluted and too complicated for my tastes.
In all fairness, the iPhone OS is more limiting in some ways than the Blackberry. The iPhone is about simplicity and elegance while the Blackberry is a swiss army knife. With a Blackberry, you can find pretty much any function you want buried in those endless menus. But they are both capable smartphones. It's up to the end user to figure out what he needs.
Anyway, I'm able to use my iPhone for business very effectively, and I do see at least a few others in my field doing the same. Plus I can use it for fun things like games and music. I really don't care if some people think that's odd.