This is an age old argument that always gets people riled up. What a lot of people tend to dismiss is the word photography / photographer is a broad term that encompasses too many levels, styles and techniques. Likewise most people here are arguing the point using similar broad terms to qualify their logic (writer, driver, artist, etc.)
Let's see how this argument holds up; "I'm here to admit that I firmly believe there is a difference between a witter and someone who just authors a book. One cannot simply author a single great book and consider themselves a writer, you have to be able to repeat the process again and again. Then, fully realizing what you have done before, apply your past experience to what you are aiming to do with the next book, and create something new."
How about this one; "I'm here to admit that I firmly believe there is a difference between an artist and someone who just draws an illustration. One cannot simply draw a single great illustration and consider themselves an artist, you have to be able to repeat the process again and again. Then, fully realizing what you have done before, apply your past experience to what you are aiming to do with the next drawing, and create something new."
The inherent problem with all these arguments is that it is forcing a singular definition onto a broad term while at the same time dismissing any other possibility. The argument falls apart not by what it is trying to define but by what it is excluding.
The definition of a photographer is "a person who takes photographs, especially one who practices photography professionally." However, the definition does not make the term exclusive to a photography professional. The other part of the argument being overlooked is that the etymology of the term is currently being redefined. Language is a living structure and changes according to social needs and changing technologies. Today's technology of image making has allowed everyone to be able to capture images digitally, therefore, by the true sense of the term, everyone is a photographer, one who takes photographs. Where the differences lie is in the qualifiers attached to the broad term. Art photographer, amateur photographer, casual photographer. How about good photography or bad photography.
The camera is a tool, much like a pencil is a tool and a brush is a tool. Owning these things does not inherently make one a photographer, writer or artist. However, once those tools are used, regardless of skill level, to create something (usually of value to someone) then that person becomes a photographer, writer or artist. Again, qualifiers will let others know to what skill level or discipline they are within those vocations.
Need some examples? the previous example of Van Gogh was a great one. Vivian Maier would be another. She was a nanny. She took "snapshots" (according to previously defined terms within this thread) that were tossed into a shoebox never to see the light of day. After her death they were discovered and now she is lauded as a great photographer. The thing is, by virtue of the definition, she was a photographer when she created those "snapshots", not a nanny. Being a nanny defined her job when tending her employer's children. The act of using a camera to record an image gave her the very broad and very loose title of photographer. However, she was not a pro photographer. She was not a photojournalist or an art photographer. By another definition she was a documentarian who utilized a camera (the tool) to record her surroundings and the people she encountered, ergo a street photographer. It's the qualifier that defines her work.
Let me backtrack here. Above I mentioned that one acquires a term (photographer, writer, artist) when one utilizes a specific set of tools and skills to create something of value. This is generally understood within the professional settings as the item of value is often of a monetary nature, however, value isn't always measured in legal tender. Ask any parent who has ever received a squiggly lined drawing from their child if that drawing has any value. Or that lumpy clay ashtray from art class, or those glued on maccarrone shell picture frames or paper mache balloon sculptures.
Ask any hobbyist photographer if their images, hanging on their wall or sitting in their computer, never seeing a dime of income don't hold any value. Or the mom or dad who take a quick, blurry image with their cell phones of their toddler at a playground. Value is subjective. Even if we don't see it that doesn't mean it's not there. Sure, that blurry picture of a five year old sliding down a slide may not garner any awards but it doesn't disqualify the person who takes a photograph as being the true definition of the word, photographer. It just makes them a lazy, undisciplined, utilitarian, observational, reflexive, one of many photographer.