fix(split-pane): gate the Split button to desktop, make it per-device

Ark0N's PR #453 review: nothing gated this feature to desktop even
though the design called for it (two 240px min-width panes plus the
divider need ~486px, and the divider has no touch handlers), and
showSplitButton was a SYNCED setting, so turning it on at a desk also
put the button in the phone header.

- Hard-hide .btn-split on phones in mobile.css regardless of the
  setting, matching the other desktop-oriented header buttons in the
  same @media (max-width: 599px) block.
- Move showSplitButton into settings-ui.js's per-device displayKeys
  set and drop it from SettingsUpdateSchema entirely, matching the
  showFileViewerButton/skin precedent (CLAUDE.md's "per-device keys
  ... must NOT be added to SettingsUpdateSchema" rule) — a desktop
  opt-in must never sync onto a phone that never asked for it. Removes
  the now-invalid server-round-trip test for the setting.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

src/web/public/settings-ui.js

@@ -3373,6 +3373,7 @@ Object.assign(CodemanApp.prototype, {
           'showTabDetachButton',
           'mobileOverviewEnabled',
           'sessionLineageLines',
+          'showSplitButton',
         ]);
         // The plan-usage chip is a PER-DEVICE display setting (desktop default ON,
         // handheld default OFF): desktop can show it while mobile stays hidden. Drop