This isn't really a taxonomy of photographs, it's really more of an incomplete list of overlapping categories. Still, I think there's some use in it.
Over Processed Crap
You see a lot of this on flickr. This is all about applying post-processing to a mediocre image to create something that is eye-catching. HDR, over-sharpening, over-saturation, all those things. Basically anything in flickr's Explore will fit in here. These are un-challenging images with visual "pop". The capture the eye, they do not engage the brain. They are perfect for the social media world of 3 second attention spans.
These photographs do evoke a reaction, but that reaction is "Wow! Great HDR!" or something. The viewer reacts to and admires the processing, the look of the thing, before the subject, and before the image qua image. The latter is probably left completely unexamined, really.
These images are tests or exercises for the photographer. One might test equipment, technique, ideas about composition, that sort of thing. Done well, these things are emotionally dead photographs that nonetheless do some things very well. One might even wonder why it feels so dead, given that the lighting is so great, or the tones are so well-placed, or the model is posed and framed so beautifully. Sometimes a technical exercise is also a fine photograph, but the point is that it's not made to be one.
Snapshots are photographic proof that the photographer was somewhere, or did something. By intention, these photographs probably evoke nothing for anyone but the photographer or other people who were there. For the photographer, the image brings back Grandma's birthday party, dinner at that one place, my daughter when she was 6 months old. See also stupid pictures of My Meal. Snapshots are the most common photograph made, by far, and serve an excellent purpose. There's nothing wrong with making snapshots.
It's sort of rude to inflict them on other people, though, since your snapshots mean nothing to me.
These photos are usually a cool angle, a reflection, an unusual viewpoint. The photographer was clearly struck by some cool thing, and took a picture of it. The photograph is all about the cool thing. Ideally, such an image evokes a "wow, what a cool thing" response, but generally no more.
The Eiffel tower reflected in the rear view mirror of a BMW. These overlap with snapshots a lot, obviously, but they may reach out a little past the photographer and reach unrelated viewers, albeit superficially. "Wow, what a cool thing!" is more of a response than "Who is the old lady blowing out the candles on the cake?".
A photograph that evokes something more might count as "art". Ansel Adams' work evokes a sense of being there, like a snapshot, but for other people. His photographs have arguably altered the the way we see mountains -- they certainly do not look the way he shot them, but they feel like it. Walker Evans shows us some sort of essence of poverty, in some of his work. Dorothea Lange's "Migrant Mother" pokes a bunch of buttons for many of us. As an aside: Ansel Adams wasn't much of an artist, but he did some art things pretty well, and he's had an immense impact on how we look at photographs and at rocks.
Art creates a reaction in the viewer. The photo shows us what was, at that instant, in front of the lens. Through myriad choices starting with what to place in front of the lens, the photographer shows us something a little new, and enlarges the viewer's mind slightly. We react not exclusively to the subject, nor to the way the image is processed, nor to any specific features of the way the image looks, but rather to all of these at once. The viewer's own preconceptions and memories and social constructs bear on the image and create some new thought or emotional complex. The eye is captured, the mind is engaged. The photograph is worth looking at more than once, and has a fair chance of being remembered for more than a few seconds.