[Backport 2.x] Add a counter to node stat api to track shard going from idle to non-idle

[ruai0511]

Description

2.x Backport PR for search idle tracking
Add a counter to node stat api to track shard going from idle to non-idle

Check List

• New functionality includes testing.
  • All tests pass
• New functionality has been documented.
  • New functionality has javadoc added
• Failing checks are inspected and point to the corresponding known issue(s) (See: Troubleshooting Failing Builds)
• Commits are signed per the DCO using --signoff
• Commit changes are listed out in CHANGELOG.md file (See: Changelog)
• Public documentation issue/PR created

By submitting this pull request, I confirm that my contribution is made under the terms of the Apache 2.0 license.
For more information on following Developer Certificate of Origin and signing off your commits, please check here.

[github-actions]

Compatibility status:

Checks if related components are compatible with change d4ba4de

Incompatible components

Skipped components

Compatible components

Compatible components: [https://github.com/opensearch-project/custom-codecs.git, https://github.com/opensearch-project/flow-framework.git, https://github.com/opensearch-project/neural-search.git, https://github.com/opensearch-project/cross-cluster-replication.git, https://github.com/opensearch-project/opensearch-oci-object-storage.git, https://github.com/opensearch-project/job-scheduler.git, https://github.com/opensearch-project/security-analytics.git, https://github.com/opensearch-project/observability.git, https://github.com/opensearch-project/geospatial.git, https://github.com/opensearch-project/k-nn.git, https://github.com/opensearch-project/notifications.git, https://github.com/opensearch-project/asynchronous-search.git, https://github.com/opensearch-project/reporting.git, https://github.com/opensearch-project/ml-commons.git, https://github.com/opensearch-project/sql.git, https://github.com/opensearch-project/common-utils.git, https://github.com/opensearch-project/anomaly-detection.git, https://github.com/opensearch-project/index-management.git, https://github.com/opensearch-project/alerting.git, https://github.com/opensearch-project/security.git, https://github.com/opensearch-project/performance-analyzer-rca.git, https://github.com/opensearch-project/performance-analyzer.git]

[github-actions]

❌ Gradle check result for 5839e47: FAILURE

Please examine the workflow log, locate, and copy-paste the failure(s) below, then iterate to green. Is the failure a flaky test unrelated to your change?

[github-actions]

❌ Gradle check result for d4ba4de: FAILURE

Please examine the workflow log, locate, and copy-paste the failure(s) below, then iterate to green. Is the failure a flaky test unrelated to your change?