### Elasticsearch Version

8.8.1+ but probably all versions

### Installed Plugins

_No response_

### Java Version

_bundled_

### OS Version

cloud

### Problem Description

This is a follow-up of [#97370](https://github.com/elastic/elasticsearch/issues/97370). 

In a case where disk spaces are reaching limits, if ILM suddenly tries to move a lot of hot shards to cold/frozen shards, it can tip the cluster health to RED because a lot of cold/frozen shards may be created and stay unassigned as long as there is not enough disk space. However, this is not a suitable indicator for service unavailability, since the data is still safely in the respective hot shards that are becoming cold/frozen, and will remain there until the cold/frozen shards are assigned so that searches can still search them.

Thus the [Health Report API](https://www.elastic.co/guide/en/elasticsearch/reference/8.8/health-api.html) should not return RED for unassigned cold/frozen shards of ILM moves.

### Steps to Reproduce

In a case where disk spaces are reaching limits, if ILM suddenly tries to move a lot of hot shards to cold/frozen shards, it can tip the cluster health to RED because a lot of cold/frozen shards may be created and stay unassigned (as long as there is not enough disk space).

### Logs (if relevant)

_No response_
server/src/main/java/org/elasticsearch/cluster/routing/allocation/ShardsAvailabilityHealthIndicatorService.java
"restarting_primaries",
                        primaries.unassigned_restarting,
                        "started_primaries",
                        primaries.started + primaries.relocating,
                        "unassigned_replicas",
                        replicas.unassigned,
                        "initializing_replicas",
                        replicas.initializing,
                        "creating_replicas",
                        replicas.unassigned_new,
                        "restarting_replicas",
                        replicas.unassigned_restarting,
                        "started_replicas",
                        replicas.started + replicas.relocating
                    )
                );
            } else {
                return HealthIndicatorDetails.EMPTY;
            }
        }
        public List<HealthIndicatorImpact> getImpacts() {
            final List<HealthIndicatorImpact> impacts = new ArrayList<>();
            if (primaries.indicesWithUnavailableShards.isEmpty() == false) {
                String impactDescription = String.format(
                    Locale.ROOT,
                    "Cannot add data to %d %s [%s]. Searches might return incomplete results.",
                    primaries.indicesWithUnavailableShards.size(),
                    primaries.indicesWithUnavailableShards.size() == 1 ? "index" : "indices",
                    getTruncatedIndices(primaries.indicesWithUnavailableShards, clusterMetadata)
                );