Return (emit) @timestamp date value in runtime field

That is exactly what I showed you... confused... you have to name it as a new field..

You add a new field...this is in the Index Pattern if you which will work for visualizations.

BUT if you actually want to do searches / aggregation etc.. etc.. you need to add it to the mapping

PUT my-index-000001/
{
  "mappings": {
    "runtime": {
      "date_month_MMM": {
        "type": "keyword",
        "script": {
          "source": "emit(doc['@timestamp'].value.month.getDisplayName(TextStyle.SHORT, Locale.ROOT));"
        }
      }
    }