[8.6] [Enterprise Search] Fix logic for long running and orphaned jobs (

#145714) (#146514)

# Backport

This will backport the following commits from `main` to `8.6`:
- [[Enterprise Search] Fix logic for long running and orphaned jobs
(#145714)](#145714)

<!--- Backport version: 8.9.7 -->

### Questions ?
Please refer to the [Backport tool
documentation](https://github.com/sqren/backport)

<!--BACKPORT [{"author":{"name":"Sander
Philipse","email":"94373878+sphilipse@users.noreply.github.com"},"sourceCommit":{"committedDate":"2022-11-22T13:27:46Z","message":"[Enterprise
Search] Fix logic for long running and orphaned jobs (#145714)\n\nThis
fixes the logic for long-running connector sync jobs: they
should\r\nlook at started_at rather than last_seen.\r\nIt also fixes the
logic for orphaned jobs, which were displaying the\r\nopposite of what
they should have been
displaying.","sha":"2e178f8f450ad6019903af17881c8368806816c8","branchLabelMapping":{"^v8.7.0$":"main","^v(\\d+).(\\d+).\\d+$":"$1.$2"}},"sourcePullRequest":{"labels":["release_note:skip","Team:EnterpriseSearch","v8.6.0","v8.7.0"],"number":145714,"url":"https://github.com/elastic/kibana/pull/145714","mergeCommit":{"message":"[Enterprise
Search] Fix logic for long running and orphaned jobs (#145714)\n\nThis
fixes the logic for long-running connector sync jobs: they
should\r\nlook at started_at rather than last_seen.\r\nIt also fixes the
logic for orphaned jobs, which were displaying the\r\nopposite of what
they should have been
displaying.","sha":"2e178f8f450ad6019903af17881c8368806816c8"}},"sourceBranch":"main","suggestedTargetBranches":["8.6"],"targetPullRequestStates":[{"branch":"8.6","label":"v8.6.0","labelRegex":"^v(\\d+).(\\d+).\\d+$","isSourceBranch":false,"state":"NOT_CREATED"},{"branch":"main","label":"v8.7.0","labelRegex":"^v8.7.0$","isSourceBranch":true,"state":"MERGED","url":"https://github.com/elastic/kibana/pull/145714","number":145714,"mergeCommit":{"message":"[Enterprise
Search] Fix logic for long running and orphaned jobs (#145714)\n\nThis
fixes the logic for long-running connector sync jobs: they
should\r\nlook at started_at rather than last_seen.\r\nIt also fixes the
logic for orphaned jobs, which were displaying the\r\nopposite of what
they should have been
displaying.","sha":"2e178f8f450ad6019903af17881c8368806816c8"}}]}]
BACKPORT-->

x-pack/plugins/enterprise_search/server/lib/stats/get_sync_jobs.ts

@@ -24,8 +24,14 @@ export const fetchSyncJobsStats = async (client: IScopedClusterClient): Promise<
   const orphanedJobsCountResponse = await client.asCurrentUser.count({
     index: CONNECTORS_JOBS_INDEX,
     query: {
-      terms: {
-        'connector.id': ids,
+      bool: {
+        must_not: [
+          {
+            terms: {
+              'connector.id': ids,
+            },
+          },
+        ],
       },
     },
   });
@@ -51,7 +57,7 @@ export const fetchSyncJobsStats = async (client: IScopedClusterClient): Promise<
           },
           {
             range: {
-              last_seen: {
+              started_at: {
                 lt: moment().subtract(1, 'day').toISOString(),
               },
             },
@@ -62,10 +68,10 @@ export const fetchSyncJobsStats = async (client: IScopedClusterClient): Promise<
   });
 
   const errorResponse = await client.asCurrentUser.count({
-    index: CONNECTORS_JOBS_INDEX,
+    index: CONNECTORS_INDEX,
     query: {
       term: {
-        status: SyncStatus.ERROR,
+        last_sync_status: SyncStatus.ERROR,
       },
     },
   });
@@ -109,7 +115,7 @@ export const fetchSyncJobsStats = async (client: IScopedClusterClient): Promise<
           {
             range: {
               last_seen: {
-                gt: moment().subtract(30, 'minutes').toISOString(),
+                lt: moment().subtract(30, 'minutes').toISOString(),
               },
             },
           },