Chrome browser selection improved with persisted hints and clearer prompts

What

• A new control-protocol message, set_chrome_browser_hints, lets a remote or gateway host tell the built-in Chrome browser extension (used for browser automation) which device to prefer on a given machine, including a preferred device ID, a list of local device IDs, and the host platform. This is stored per remote session.
• A new browserHints state (preferred device ID and local device IDs) is threaded through the extension-pairing and device-detection logic, powering a rewritten selection helper that automatically picks a persisted device, the only connected extension, or the sole local candidate, or otherwise asks the user to choose.
• When asking the user to pick a browser, Claude Code now groups the choices into "Browsers on this computer" and "Other connected browsers," always offers a "switch to a different browser" option, and is instructed to always ask when the choice is ambiguous.
• The list_connected_browsers tool now reports an onThisComputer flag (a more reliable signal than the previous "is local" heuristic) and an inUse flag marking which browser the session's actions currently target, plus readable platform names (macOS, Windows, Linux, etc.) instead of raw platform strings.
• New helper text tells Claude when a browser was auto-selected (no need to ask the user) versus when it must ask, and a new reportBrowserSelected telemetry event records the outcome along with how many browsers were connected and how many were local.
• A new timeout message for unresponsive tool calls now mentions that Chrome might be asleep on another computer, depending on whether a routing acknowledgment was received.

Why With several Chrome browsers or computers potentially connected, these changes reduce unnecessary prompts by remembering and auto-selecting the right browser when possible, while making it clearer which browser is in use and giving better options when a manual choice is genuinely needed.