A server that can store extensible schema and communicate using SQL for describing transactions to be performed is commonly called a:

A server that can store extensible schema and communicate using SQL for describing transactions is most accurately referred to as a database server. Database servers are designed to manage databases and offer functionality to perform tasks such as storing, retrieving, and manipulating data through SQL (Structured Query Language). This accommodation includes handling data in a structured format and allowing multiple transactions, making them crucial for applications that require data management.

In contrast, directory servers focus on storing and retrieving directory information, which is typically hierarchical and may not utilize SQL or handle transactional data in the same way as a database server. Web servers are primarily responsible for serving web content and providing web services rather than directly managing databases. Messaging servers are designed to facilitate communication between applications by managing message queues and ensuring message flow; they do not primarily handle data storage and querying with SQL.

Thus, the ability to store extensible schema and manage transactions using SQL distinctly classifies the server as a database server.