Is there a way to display the currently open Lister in the toolbar?
Not in the toolbar, but you can display it in the titlebar, if you mean the layout name.
Yes, I would like to display the layout name of the opened Lister layout in the titlebar. How can I do that please?
There's checkbox at the bottom of Preferences / Display / Options:
- Display layout name (if any)
If you are looking for more options this addon script might be of interest.