[FEATURE]: Option to show project/directory name in session tabs (horizontal and vertical) #56749

Feature hasn't been suggested before

• I have verified this feature I'm about to request hasn't been suggested before.

Describe the enhancement you want to request

With tabs.scope set to global, sessions from several working directories share one tab strip, and the only label on a tab is the session title. Titles come from the first message, so two sessions in different repos often read the same ("Fix login bug"). I can't tell which directory a tab belongs to until I switch to it. This is the main reason I used to run one instance per directory inside tmux.

I saw that vertical tabs already have a second line with the project name (sessionTabDetail). I'd like something like that to be available as a setting that works in both layouts:

• A cli.json option under tabs (name is up to you, e.g. tabs.label) that adds the project or directory name to the tab.
• It applies to horizontal tabs as well as vertical ones. Horizontal tabs have no detail line, so a name: title style prefix would probably be the simplest form there.
• Display only. The stored session title stays untouched.
• Whether it's on by default is your call. I'd just like to be able to turn it on.

Name vs. full path (or ~-shortened) could be a second value if that's cheap, since two directories with the same basename are still ambiguous.

For now I'm using a small local plugin that renames the session to dir:title on session.created / session.renamed. It works, but it writes the prefix into the session data, so it also shows up in the session list, exports and the API. A display-level option in the tab strip would avoid that, and a plugin can't do it today because there is no slot for tab titles.

Related: #50228, #51759.