While this sounds like a great addition, I think it's a little too involved. When people are playing, the last thing they need to do when trying to read chat is havng to click extra buttons/look at which button to click. Players are on the go and don't have much time to type in between rounds already. Something simple such as pressing [insert button here] to have a much larger chat box pop up is easier, smoother, and less steps to achieve the same desire.
Plus this just reminds me of what's already in game; the ability to mute/ignore the different chat options. I'm aware that you're coming at it from an organizing way and not the ignoring option.
I just don't see the demand for a more organized chat box over the demand for an option to enlarge the chatbox.