feat(badge): show the multiplexer a session runs in next to the terminal

[mutoe]

Problem

A CLI running under tmux inside iTerm2 is labelled just iTerm2. The multiplexer it actually sits in is invisible on the badge, even though several sessions in the same terminal may live in different tmux panes — or different tmux sessions entirely.

The data is already there. From a real ~/.codeisland/sessions.json:

{"source":"claude", "termApp":"tmux", "termBundleId":"com.googlecode.iterm2",
 "tmuxEnv":"/private/tmp/tmux-501/default,27935,0", "tmuxPane":"%11", ...}

termApp is tmux, and tmuxEnv / tmuxPane are captured alongside it — activateTmux already relies on them for click-to-jump. But terminalName resolves from termBundleId alone, so it hits com.googlecode.iterm2 and returns "iTerm2"; the multiplexer layer never reaches the badge. Nothing in the collection path needs to change.

Change

SessionSnapshot.multiplexerLabel — the multiplexer layered on top of the terminal — rendered by TerminalBadge as a small chip after the terminal name, in the chip vocabulary already used by the queue counter and the AskUserQuestion header. Both the local and remote branches of the badge get it: SSH sessions are frequently inside tmux too.

The chip augments the terminal name, never replaces it — you still want to know it's iTerm2.

Two deliberate exclusions

cmux is not reported as a multiplexer. It is a terminal app, already named by terminalName (if lower.contains("cmux") { return "cmux" }). Reporting it again here would render the same word twice on one badge. There's a regression test pinning this.

screen is not covered. STY is never captured anywhere in the codebase, so supporting it means touching the hook collection path — a separate change. tmux and zellij both have their env vars already recorded, so they cost nothing.

Nesting

Nested multiplexers each leave their env vars behind, so multiplexerLabel returns the innermost layer — the one the CLI actually sits in. zellij inside tmux reads zellij. This matches how click-to-jump already treats the pair (see the zellij-inside-Terax handling in v1.0.31).

Tests

New SessionMultiplexerLabelTests, built test-first:

• tmux inside iTerm2 → "tmux", and terminalName still "iTerm2" (the chip must not swallow the terminal)
• zellij → "zellij"
• zellij nested in tmux → "zellij" (innermost wins)
• cmux → nil, with terminalName asserted as "cmux" (pins the no-double-render rule)
• plain terminal → nil

Full suite green locally on macOS 26.4 / Xcode 26 (Swift 6.3.3): 890 tests, 2 skipped, 0 failures. swift build clean.

Note

Independent of #325 — different files apart from a non-overlapping region of NotchPanelView.swift, and either can merge first.

[mutoe]