If someone is writing in pictographs, would the correct verb to describe the action be, "to write"? Or would it be to depict or something along those lines?
A single pictograph is a depiction, but I’d say that a sequence of pictographs (emoji, &c.) constitutes writing. There are many languages whose writing systems were originally pictographic, such as Chinese, Ancient Egyptian, and Mayan hieroglyphs, and we think of these as writing in the usual sense.
One writes language, whereas one depicts reality. If the product makes use of or otherwise appeals to a structured linguistic system, then it's writing -- if not, it's depiction.