Zero:     "not to have them added as text as is usual in a terminal window"
taleinat: "removing prompts from the shell window's text widget"

Zero:     "print the values of ps1 and ps2 in the proposed ShellIO subclass"
taleinat: "separate sidebar showing where prompts and user input were"

We appear to be in agreement.

terry.reedy: "Labels, such as '>>>', 'Out', 'Inp', and 'Err' would be used"
Zero:        "Having IDLE react to sys.ps1 and sys.ps2"

My suggestion is to take those labels terry.reedy talks about from the values of ps1 and ps2 since they are already documented and standard for "the interpreter ... in interactive mode." If psX needs to introduced for other prompts that may be needed ("Maybe use 'Con', maybe use Dbg and Wrn."), it would provide a sensible way to customize those labels as well.