This is my understanding: “-graphy” is writing or drawing — I like to say painting since it’s more romantic — and “photo” is light; literally the photons that produce the spectrums of visible, IR and UV light disseminated by our eyes and brains. “Photography” is therefore painting (or writing or drawing) with light.
There is stills photography, and motion photography which can be for cinema or video; we say videography and cinematography, respectively. The difference between them is a videographer captures moments whereas a cinematographer creates moments.
The director of photography is an upper-middle-level management position on a film or television set, responsible for overseeing several different departments related to creating the image on screen. The DP is mostly concerned with the organization of those departments, collaborating with the director/s, art director/production designer, and producer/s with the visual tone of the work, and supervising the execution of that work via the team members of each department underneath the DP. The DP usually sits with the director in video village and rarely operates or digs in with the crew while the cameras are rolling.
It is my understanding that DPs who tend to be entrenched with the camera crew and personally operate the A camera prefer to be called cinematographers on set, though they are cinematographers by artistic profession and choice whereas “director of photography” is merely a title or honorific.
I almost always operate, and almost always gaffe, and prefer the term cinematographer, but I will always turn my head and smile when I hear someone affectionately refer to me as DP.