[8.10] [Search] Update connector also updates status (#165983) (#166059)

# Backport

This will backport the following commits from `main` to `8.10`:
- [[Search] Update connector also updates status
(#165983)](https://github.com/elastic/kibana/pull/165983)

<!--- Backport version: 8.9.8 -->

### 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":"2023-09-08T09:14:17Z","message":"[Search]
Update connector also updates status (#165983)\n\n##
Summary\r\n\r\nConverting a connector from native to client now updates
the
connector\r\nstatus.","sha":"9deae84801fa770a41d107c633d3b6e356631b03","branchLabelMapping":{"^v8.11.0$":"main","^v(\\d+).(\\d+).\\d+$":"$1.$2"}},"sourcePullRequest":{"labels":["release_note:skip","Team:EnterpriseSearch","v8.10.0","v8.11.0"],"number":165983,"url":"https://github.com/elastic/kibana/pull/165983","mergeCommit":{"message":"[Search]
Update connector also updates status (#165983)\n\n##
Summary\r\n\r\nConverting a connector from native to client now updates
the
connector\r\nstatus.","sha":"9deae84801fa770a41d107c633d3b6e356631b03"}},"sourceBranch":"main","suggestedTargetBranches":["8.10"],"targetPullRequestStates":[{"branch":"8.10","label":"v8.10.0","labelRegex":"^v(\\d+).(\\d+).\\d+$","isSourceBranch":false,"state":"NOT_CREATED"},{"branch":"main","label":"v8.11.0","labelRegex":"^v8.11.0$","isSourceBranch":true,"state":"MERGED","url":"https://github.com/elastic/kibana/pull/165983","number":165983,"mergeCommit":{"message":"[Search]
Update connector also updates status (#165983)\n\n##
Summary\r\n\r\nConverting a connector from native to client now updates
the
connector\r\nstatus.","sha":"9deae84801fa770a41d107c633d3b6e356631b03"}}]}]
BACKPORT-->

x-pack/plugins/enterprise_search/server/lib/connectors/put_update_native.ts

@@ -18,6 +18,7 @@ export const putUpdateNative = async (
   const result = await client.asCurrentUser.update<Connector>({
     doc: {
       is_native: isNative,
+      status: ConnectorStatus.CONFIGURED,
     },
     id: connectorId,
     index: CONNECTORS_INDEX,