How about just appending an extra word at the beginning of the word 'photographer', which would help people understand that the broader term of 'photographer' applies to many different types. Then you could have 'amateur photographer', or 'professional photographer', 'wedding photographer', 'artistic photographer', 'product photographer', 'nature photographer', etc. All are photographers - some make money at it, some have incredible artistic capabilities, some are hobbyists, some are beginners...but all photographers.
What Tim said is very good - and some other points added are all true - a photographer can probably be defined by many of those methods. Certainly, if you feel the need to ask if you are a photographer, you probably are!
I think that your interest in something defines you as a practicer of it. I've had similar debates about 'drivers' - the generic term being 'one who drives'. But to me, many people who drive are not 'drivers' - they lack the emotional connection to the process. I think you have to have some emotional connection to a practice to be considered a practitioner of it, and 'photographer' falls into that definition. Any person with an emotional connection to photography would be a photographer in my mind. If the camera is nothing more than an appliance with which you complete a particular task (taking a picture), and the purpose of taking the picture is for purely documenting and filing for later reference...you are probably not a photographer (my mother, for example - she takes pictures on her vacations, but only a handful, usually involving specific people or moments that she felt compelled to document and file in an album for future reference).
I'm a photographer. There is no doubt. Because I care about photography, and feel emotionally connected to it. I was a photographer when I first picked up a camera, and through all of those horrible snapshots I took along the way...no matter how bad my photos were, I always enjoyed taking them, tried to learn more and get better, and when viewing pictures I think just as much about how it was taken, the camera settings, and the conditions in the photo as I do about the subject. Over the past few years, I have been hired for two shoots, shot a model portfolio for a friend, and sold a handful of photos (unsolicited), but I don't classify myself as a professional photographer...just a growing amateur nature/landscape/Disney/fashion/wildlife/vacation photographer!