change docs for rm name to cluster name

docs/reference/deploy/master-config-reference.rst

@@ -251,7 +251,7 @@ specified under the primary resource_manager key here is considered the default.
 ================
 
 Optional for single resource manager configurations. Required for multiple resource manager
-(Multi-RM) configurations. Specifies the resource manager'sassociated cluster name. This references
+(Multi-RM) configurations. Specifies the resource manager's associated cluster name. This references
 the cluster on which a Determined deployment is running. Defaults to ``default`` if not specified.
 For Kubernetes installations with additional resource managers, ensure unique names for all resource
 managers in the cluster.
@@ -1221,7 +1221,8 @@ workloads that require proxying on remote clusters is under development.
 To define a single resource manager or designate the default resource manager, do not define it
 under ``additional_resource_manager``; instead, use the primary ``resource_manager`` key.
 
-Resource manager names must be unique among all defined resource managers.
+Resource managers' cluster names (``resource_manager.cluster_name``) must be unique among all
+defined resource managers.
 
 Any additional resource managers must have at least one resource pool assigned to them. These
 resource pool names must be defined and must be distinct among all resource pools across all

docs/setup-cluster/k8s/setup-multiple-resource-managers.rst

@@ -162,12 +162,12 @@ the same as the “cluster name” for a given cluster.
  WebUI
 *******
 
-In the WebUI, the resource manager name is visible for each resource pool.
+In the WebUI, the resource manager's associated cluster name is visible for each resource pool.
 
 To view resource managers:
 
 -  In the WebUI, navigate to the cluster view.
--  Each resource pool card displays **Resource Manager Name**.
+-  Each resource pool card displays **Cluster Name**.
 
 This field helps identify whether a resource pool is managed locally or by another manager, tagged
 as “Remote” if defined in the :ref:`master-config-reference` file.
@@ -177,11 +177,11 @@ as “Remote” if defined in the :ref:`master-config-reference` file.
 Let's say your environment contains a resource manager and a resource pool that have both adopted
 the name "default" due to their unnamed status, and you configure an additional resource pool named
 "additional-rm” with a resource pool named "test". When you sign in to the WebUI, you'll see both
-the default and test resource pools. The resource manager name for the default pool will display
-“default”, while the test pool displays "additional-rm" (or the name you specified).
+the default and test resource pools. The resource manager's cluster name for the default pool will
+display “default”, while the test pool displays "additional-rm" (or the name you specified).
 
 **Visibility and Access**
 
-**Resource Manager Name** is only visible to administrators or users with permissions to define
-multiple resource managers. Users can view all resource pools along with each resource pool's
-manager name to help distinguish between local and remote resource pools.
+**Cluster Name** is only visible to administrators or users with permissions to define multiple
+resource managers. Users can view all resource pools along with the cluster name of each resource
+pool's manager to help distinguish between local and remote resource pools.