As a purist, I would recommend PostgreSQL, which is the closest distribution to fully ANSI-complaint. Unfortunately the industry does not run on purists unless coding in Ruby on Rails (RoR) . The most common RoR installation is on the Heroku platform, which offers PostgreSQL as its default database. In any case, it is not heavily used as giants like Oracle or SQL Server (Microsoft) and this might make it more secure than the latter two – fewer hackers trying to break PostgreSQL or its security. Maybe this is a good reason to be open source as anyone can fix a bug before it gets maliciously exploited.
As a nostalgist, I would recommend Oracle, which is the system that I learned with. Unfortunately Oracle has somewhat of a bad reputation in the industry after taking over Sun. Some developers opted to quit rather than joining Oracle. The OpenOffice development group formed LibreOffice, while MySQL development group formed MariaDB. Oracle is proprietary (closed) source.
As a practicalist, I would recommend MariaDB and not MySQL to stay away from whatever toxic environment that has made many programmers leave Oracle. As a drop-in replacement of MySQL, MariaDB has taken over what had been the MySQL place in the LAMP (from Linux, Apache, MySQL/MariaDB, Perl/PHP/Python to Linux, Apache, MariaDB, Perl/PHP/Python) web stack maintaining the full open source mantra. Note that the web stack could also be referred to as WAMP when running on a Windows server.
LAMP is an Open Source Web development platform that uses Linux as the operating system, Apache as the Web server, MySQL as the relational database management system and PHP as the object-oriented scripting language. (Sometimes Perl or Python is used instead of PHP.)
https://www.techtarget.com/whatis/definition/LAMP-Linux-Apache-MySQL-PHP
As a realist, rather than teaching MariaDB, I would teach MySQL as it has a much stronger market share and more students might be interested in MySQL rather than its fork. In other words, new comers to the industry might have heard of MySQL, but not MariaDB. MySQL has a combination of open source (full open source for the Community Edition, which I am set to teach next) and proprietary (closed) sources.
As a minimalist, I would recommend SQLite (actually sqlite3) – no server, just data. SQLite is used in many systems including web browsers and mobile telephones. The only drawback is the lack of robust security or user authentication, but it is fully open source so its users clean up bugs right away.