It seems like you might be personificating(sp?) the computer a little bit. When you latch a fence gate closed, you wouldn't say you were communicating with the fence gate, you are just operating the mechanism.
When I say data, this could be anything, so in your example the data could be the input of a 2d painting (image) and the computer would convert that input data, into the output data of a 3d model (stl). If you are playing a video game, the data would be the representation of the world you are in, and when you manipulate your player you are just updating the data which is the representation of the world, and then displaying the updated result (the output to the screen for the player).
(some what unrelated) Although your third paragraph does cover of one my favorite thought experiment when I'm learning new things, and that is trying to find ways to take what I learned in one medium and applying it to other mediums; it's always fun to see that something I learned while programming (or any creation) can be easily translated to another medium like cooking or playing music. It's good to have many different hobbies just so you can see how all forms of creation work about the same.
It's also good because the last thing you want to do look at a screen after spending the day coding. The only time I seem to be able to stop playing video games as a hobby is when I've been elbow deep in code for the last 10 hours.