Azure SQL Database is Microsoft's fully managed, relational Platform-as-a-Service (PaaS) database. This directly matches the description of a "managed relational cloud database service."

Azure Synapse Analytics integrates what was formerly Azure SQL Data Warehouse. Its dedicated SQL pools use a massively parallel processing (MPP) architecture specifically designed to run complex queries on petabyte-scale relational data warehouses.

Azure Data Lake Analytics (now being retired, but historically correct) was an on-demand analytics job service designed to run massively parallel data transformation and processing programs (using U-SQL) across petabytes of data in a data lake.

Azure HDInsight is the managed service for open-source distributed analytics frameworks, most notably Apache Hadoop and Apache Spark. The description "An open-source framework for the distributed..." clearly points to this service.

Azure SQL Database:

Source: Microsoft Azure Documentation, "What is Azure SQL Database?"

Reference: "Azure SQL Database is a fully managed platform as a service (PaaS) database engine... SQL Database is a relational data service..."

Azure Synapse Analytics (MPP):

Source: Microsoft Azure Documentation, "Dedicated SQL pool (formerly SQL DW) architecture in Azure Synapse Analytics"

Reference: "Dedicated SQL pool (formerly SQL DW) leverages a scale-out architecture to distribute computational processing of data across multiple nodes. The unit of scale is a Data Warehouse Unit (DWU), which represents an abstract bundle of compute resources... Dedicated SQL pool (formerly SQL DW) uses a massively parallel processing (MPP) architecture..."

Azure Data Lake Analytics:

Source: Microsoft Azure Documentation, "What is Azure Data Lake Analytics?"

Reference: "Azure Data Lake Analytics is an on-demand analytics job service that simplifies big data... you can write queries to transform your data... The analytics service can handle jobs of any scale instantly..." This service was known for its massively parallel execution of U-SQL jobs.

Azure HDInsight:

Source: Microsoft Azure Documentation, "What is Azure HDInsight?"

Reference: "Azure HDInsight is a managed, full-spectrum, open-source analytics service in the cloud for enterprises. With HDInsight, you can use open-source frameworks such as Apache Spark, Apache Hadoop, Apache Kafka, Apache HBase, and Storm in your Azure environment."