Note: In this case the Tomcat server had to be restarted by cloud services (See PTC case 12796347), so the problem was actually one of stale handle to and "old" JDBC configuration. You must use the default "Database" template that comes with initial install (example 1 in the original post).
to build a Database Thing and have installed the mysql-connector-java-5137-binJDBCTemplate before restart.
(At least that's the order I performed to get it to work...)
See PTC Case CS219958 for details of installing the JDBC Connector extension and then JDBC drivers.
As of now, I have connectivity to an external MySQL database and can issue both DDL and DML.