[DOCS] Note ILM uses snapshot of user privileges (#67393) (#67401)

docs/reference/ilm/apis/put-lifecycle.asciidoc

@@ -19,9 +19,10 @@ definitions of policy components.
 
 * If the {es} {security-features} are enabled, you must have the `manage_ilm`
 cluster privilege to use this API. You must also have the `manage` index
-privilege on all indices being managed by `policy`. All operations executed by
-{ilm} for a policy are executed as the user that put the latest version of a
-policy. For more information, see <<security-privileges>>.
+privilege on all indices being managed by `policy`. {ilm-init} performs
+operations as the user who last updated the policy. {ilm-init} only has the
+<<security-privileges,privileges>> assigned to the user at the time of the
+last policy update.
 
 [[ilm-put-lifecycle-desc]]
 ==== {api-description-title}

docs/reference/ilm/ilm-index-lifecycle.asciidoc

@@ -24,6 +24,11 @@ For time series indices, you need to associate the lifecycle policy with
 the index template used to create new indices in the series. 
 When an index rolls over, a manually-applied policy isn't automatically applied to the new index.
 
+If you use {es}'s Security features, {ilm-init} performs operations as the user
+who last updated the policy. {ilm-init} only has the
+<<security-privileges,privileges>> assigned to the user at the time of the
+last policy update.
+
 [discrete]
 [[ilm-phase-transitions]]
 === Phase transitions