feat(frontend): update sessions page for enhanced health monitoring

- Add new status badges: starting, probing, unhealthy
- Show tunnel error only when tunnel_status is unreachable
- Show app error badge with status code for error_response
- Add collapsible probe output section for diagnostics
- Update health polling to check all active instances
- Only show Recreate Tunnel button for unreachable tunnels

openspec/changes/instance-health-monitoring/specs/tool-instances/spec.md

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+## MODIFIED Requirements
+
+### Requirement: Status Monitoring
+The system SHALL track tool status with startup and health states.
+
+#### Scenario: Status check with health details
+- **GIVEN** a tool instance
+- **WHEN** status is queried
+- **THEN** the real-time container status is returned:
+  - `pending`: Instance created, container not yet started
+  - `starting`: Container is running, readiness probe in progress
+  - `running`: Container is running and probe passed (or terminal tool)
+  - `unhealthy`: Container is running but probe failed/timed out
+  - `stopped`: Container was stopped by user
+  - `error`: Container failed to start or crashed
+
+## ADDED Requirements
+
+### Requirement: Health check endpoint enhancement
+The system SHALL provide detailed health information through the health check endpoint.
+
+#### Scenario: Health check with container and tunnel status
+- **GIVEN** a running instance
+- **WHEN** `GET /instances/{id}/health` is called
+- **THEN** the response includes:
+  - `healthy`: boolean - overall health
+  - `container_status`: "running", "exited", "restarting", or "not_found"
+  - `tunnel_status`: "healthy", "unreachable", "error_response", or "not_applicable"
+  - `tunnel_status_code`: HTTP status code or null
+  - `probe_status`: "passed", "failed", "pending", or "not_configured"
+  - `last_probe_output`: string or null
+
+### Requirement: Smart tunnel recreation
+The system SHALL only allow tunnel recreation when the tunnel itself is broken.
+
+#### Scenario: Recreate tunnel for unreachable tunnel
+- **GIVEN** an instance with `tunnel_status: "unreachable"`
+- **WHEN** the recreate tunnel endpoint is called
+- **THEN** the tunnel is recreated
+- **AND** the new URL is returned
+
+#### Scenario: Block recreation for application errors
+- **GIVEN** an instance with `tunnel_status: "error_response"` (e.g., HTTP 502)
+- **WHEN** the recreate tunnel endpoint is called
+- **THEN** the request is rejected with 400 Bad Request
+- **AND** the error message explains the tunnel is working but the application is returning errors
+
+## REMOVED Requirements
+
+None.