There's a little buglet in #85387 (which was the PR that closed #84046), specifically, it doesn't account for the possibility that there's both a default/request **and** and a final pipeline, and that they can both touch the `@timestamp` field.

`IngestService#postIngest` was added in #85387, and it's intended that the logic there only runs **once**, after all the pipelines associated with the request have been executed. But as it's currently written, the logic actually runs after each individual pipeline runs.

The consequence of that is that if there's a default/request and a final pipeline in use for a document, and the final pipeline changes the timestamp field, the `rawTimestamp` recorded for the `IndexRequest` associated with the document will mistakenly use the value it sees after the default/request pipeline runs, and **not** the most correct one (from after when the final pipeline runs).

/cc @martijnvg 
server/src/main/java/org/elasticsearch/ingest/IngestService.java
processorMetrics.forEach(t -> {
                Processor processor = t.v1();
                IngestMetric processorMetric = t.v2();
                statsBuilder.addProcessorMetrics(id, getProcessorName(processor), processor.getType(), processorMetric);
            });
        });
        return statsBuilder.build();
    }
    /**
     * Adds a listener that gets invoked with the current cluster state before processor factories
     * get invoked.
     * <p>
     * This is useful for components that are used by ingest processors, so that they have the opportunity to update
     * before these components get used by the ingest processor factory.
     */
    public void addIngestClusterStateListener(Consumer<ClusterState> listener) {
        ingestClusterStateListeners.add(listener);
    }
    
    static String getProcessorName(Processor processor) {
        
        if (processor instanceof ConditionalProcessor conditionalProcessor) {
            processor = conditionalProcessor.getInnerProcessor();
        }
        StringBuilder sb = new StringBuilder(5);
        sb.append(processor.getType());
        if (processor instanceof PipelineProcessor pipelineProcessor) {
            String pipelineName = pipelineProcessor.getPipelineTemplate().newInstance(Map.of()).execute();
            sb.append(":");
            sb.append(pipelineName);