Hello blubb,
Yes, it is correct that you need to delete your HDInsight cluster every time you stop working on it in order to avoid generating costs.
When you create an HDInsight cluster, you are charged for the compute and storage resources that are allocated to the cluster, regardless of whether you are actively using the cluster or not.
To avoid incurring unnecessary costs, it is recommended that you delete your HDInsight cluster when you are not actively using it. This will release the compute and storage resources that were allocated to the cluster.
Azure Storage is a separate service from HDInsight, and the data stored in Azure Storage is billed separately based on the amount of data stored and the number of transactions performed on that data.
I see the following feedback items submitted regarding starting/stopping HDInsight clusters to avoid charges:
https://feedback.azure.com/d365community/idea/c8013f73-7626-ec11-b6e6-000d3a4f032c
https://feedback.azure.com/d365community/idea/0da76cf1-7626-ec11-b6e6-000d3a4f032c
Reference document: https://learn.microsoft.com/en-us/azure/hdinsight/hdinsight-administer-use-portal-linux#pauseshut-down-clusters
I hope this answers your question. Please let me know if you have any further questions.
If this answers your question, please consider accepting the answer by hitting the Accept answer and up-vote as it helps the community look for answers to similar questions.