### 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
}
        /**
         * Returns the diagnosis for unassigned primary and replica shards.
         * @param verbose true if the diagnosis should be generated, false if they should be omitted.
         * @param maxAffectedResourcesCount the max number of affected resources to be returned as part of the diagnosis
         * @return The diagnoses list the indicator identified. Alternatively, an empty list if none were found or verbose is false.
         */
        public List<Diagnosis> getDiagnosis(boolean verbose, int maxAffectedResourcesCount) {
            if (verbose) {
                Map<Diagnosis.Definition, Set<String>> diagnosisToAffectedIndices = new HashMap<>(primaries.diagnosisDefinitions);
                replicas.diagnosisDefinitions.forEach((diagnosisDef, indicesWithReplicasUnassigned) -> {
                    Set<String> indicesWithPrimariesUnassigned = diagnosisToAffectedIndices.get(diagnosisDef);
                    if (indicesWithPrimariesUnassigned == null) {
                        diagnosisToAffectedIndices.put(diagnosisDef, indicesWithReplicasUnassigned);
                    } else {
                        indicesWithPrimariesUnassigned.addAll(indicesWithReplicasUnassigned);
                    }
                });
                if (diagnosisToAffectedIndices.isEmpty()) {
                    return List.of();
                } else {
                    return diagnosisToAffectedIndices.entrySet().stream().map(e -> {
                        List<Diagnosis.Resource> affectedResources = new ArrayList<>(1);
                        if (e.getKey().equals(ACTION_RESTORE_FROM_SNAPSHOT)) {
                            Set<String> restoreFromSnapshotIndices = e.getValue();
                            if (restoreFromSnapshotIndices != null && restoreFromSnapshotIndices.isEmpty() == false) {
                                affectedResources = getRestoreFromSnapshotAffectedResources(
                                    clusterMetadata,
                                    systemIndices,
                                    restoreFromSnapshotIndices,