Using a second monitor for live streaming is more useful for things useful to the streamer but which do not need to be shown on screen to readers (e.g. frantic Stack Overflow research when the code breaks, or analytics as the article mentions).

While it may not work for everyone, Casey Muratori (of the Handmade Hero referenced in the article) shows pretty much his whole desktop, including the times he is looking up documentation. Granted, he's not searching SO, but actual API documentation for Windows OpenGL or CPU instructions, he goes through his whole thought process including the time and effort it takes to understand and interpret documentation.

Then again, he also doesn't interact with chat at all while programming and only looks back at the end and answers questions afterward. So he breaks a few pieces of advice others might say are essential for a stream.

I think showing your process is really important and helpful to those watching your stream. I always show what I'm looking up in my browser. Folks at home are really great at helping me read it, as I'm often talking while trying to skim the docs quickly - not a good combination!

I don't know much about streaming, but having watched lots of twitch channels, I think some streamers use one screen for stream display, and the other for everything they want to see but don't want the viewers to see e.g. chat, sensitive stuff, etc.

That's exactly what I do when streaming Overwatch, especially when my friend talks to me via IRC and I reply with voice. That said my monitor is big enough to have both the game and IRC visible, and I could set OBS just to record the game.