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/instance-startup-health/spec.md

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+## ADDED Requirements
+
+### Requirement: Container startup verification
+The system SHALL verify that containers reach a running state before marking instances as "running".
+
+#### Scenario: Container starts successfully
+- **WHEN** `docker compose up` completes
+- **THEN** the system polls `docker ps` every 2 seconds for up to 30 seconds
+- **AND** when the container state is "running", the instance status becomes "starting"
+- **AND** the readiness probe begins execution
+
+#### Scenario: Container fails to start
+- **WHEN** `docker compose up` completes
+- **AND** the container exits within 30 seconds
+- **THEN** the instance status becomes "error"
+- **AND** the container exit code is stored in the error message
+
+#### Scenario: Container stays in restarting loop
+- **WHEN** `docker compose up` completes
+- **AND** the container remains in "restarting" state after 30 seconds
+- **THEN** the instance status becomes "error"
+- **AND** the error message indicates the container is stuck restarting
+
+### Requirement: Readiness probe execution
+The system SHALL execute readiness probes for web-enabled tool instances before marking them as "running".
+
+#### Scenario: Probe succeeds
+- **GIVEN** a tool instance with status "starting"
+- **AND** the tool type has a readiness probe configured
+- **WHEN** the probe command returns exit code 0 within the timeout
+- **THEN** the instance status becomes "running"
+- **AND** the tunnel is created (for web tools)
+
+#### Scenario: Probe times out
+- **GIVEN** a tool instance with status "starting"
+- **AND** the tool type has a readiness probe configured
+- **WHEN** the probe does not succeed within the configured timeout (default 30s)
+- **THEN** the instance status becomes "unhealthy"
+- **AND** the tunnel is still created (the container is running)
+- **AND** the last probe output is stored for diagnostics
+
+#### Scenario: Terminal tool skips probe
+- **GIVEN** a tool instance for a terminal-only tool type
+- **WHEN** the container reaches "running" state
+- **THEN** the instance status immediately becomes "running"
+- **AND** no readiness probe is executed
+
+### Requirement: Container health monitoring
+The system SHALL check container health in addition to tunnel health.
+
+#### Scenario: Container is healthy
+- **GIVEN** a running instance
+- **WHEN** the health endpoint is queried
+- **THEN** the response includes `container_status: "running"`
+- **AND** the response includes `container_health: "healthy"` if Docker healthcheck exists
+
+#### Scenario: Container has crashed
+- **GIVEN** a running instance
+- **WHEN** the container exits or is stopped externally
+- **AND** the health endpoint is queried
+- **THEN** the response includes `container_status: "exited"`
+- **AND** the response includes `healthy: false`
+- **AND** the instance status in the database is updated to "error"
+
+## 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
+
+## REMOVED Requirements
+
+None.