Fix ILM start and stop doc tests (#46559)

This uses whatever the server retrieves, rather than hardcoded
"STOPPING" and "STOPPED" since the server may go to STOPPED before the
request is issued.

Resolves #46528

docs/reference/ilm/start-stop-ilm.asciidoc

@@ -107,6 +107,7 @@ GET _ilm/status
   "operation_mode": "STOPPING"
 }
 --------------------------------------------------
+// TESTRESPONSE[s/"STOPPING"/$body.operation_mode/]
 
 The ILM service will then, asynchronously, run all policies to a point
 where it is safe to stop. After ILM verifies that it is safe, it will
@@ -128,6 +129,7 @@ GET _ilm/status
   "operation_mode": "STOPPED"
 }
 --------------------------------------------------
+// TESTRESPONSE[s/"STOPPED"/$body.operation_mode/]
 
 [float]
 === Starting ILM