I have 2 desks in my office, both with multiple screen layouts. Your question made me think about how I use them and it comes down to the task I am performing.
Like others, when I am programming I typically have an IDE where I am doing work on one and a reference open on another. When doing web development my third monitor usually has a browser where I can immediately refresh my work to see results, for other development it may be a virtual machine or remote desktop that I am logged into.
When I am doing academic work, I often have EndNote (reference manager) on one monitor, the document I am writing on another and the documents I am finding / reading on the third.
Since both my desks are next to each other, I often "borrow" a monitor from the other setup to keep communication windows open (Skype, Lync, Hangouts, #Slack etc.) This allows me to keep in touch with coworkers and colleagues without having to flip windows every time I get a message.
So I would say there are three purposes identified:
With your background in web development have you read things like Krug's Don't Make Me Think and William's The Non-Designer's Design Book? These are focused more on the design aspect of web however they contain some good underlying principles for data visualization as well.
Tufte's book are all great for underlying principles even though, as you noted, they aren't focused on modern technologies. Beautiful Evidence from 2006 has some updated thoughts but he still borrows heavily from his earlier books.
For general multimedia concepts, Mayer's Multimedia Learning is good from a human learning perspective (my background).
I found Data Points: Visualization That Means Something to be a good modern guide.
From my perspective, I am glad you are looking down the road and recognizing that after the data are analyzed the analysis must be communicated.
You are definitely right in that we need to think about how it will look to another human being.
If you are interested in pursuing this idea further, Don Norman has written a number of books about design in general. These are not about graphic design but just design thinking. The Psychology of Everyday Things is a classic and Emotional Design builds on the work of people like Antonio Damasio with regard to the role of emotion in cognition. Norman has another book called The Design of Everyday Things which I have not read but I imagine is a great read as well.
All of these works emphasize the role of design in helping humans accomplish their goals. Some practitioners of data analytics view the output of prose, charts, tables and graphs as the final product. In most cases however the final product of a data analytics effort is a decision. That decision might be to do more research, to buy one company versus another or propose a new policy to Congress. Regardless of the nature of the decision, how well you design the output will have an impact on the quality of the decision made.