The ranking of log levels is exactly as pictured in that dropdown - trace < debug < info < warn < error < fatal (which we never use in Ignition).
If you log a message in your script, it is logged to the gateway system log (per the logback.xml file in the gateway install directory) as well as output to stdout, which is captured by the 'service wrapper' process and dumped to the wrapper.log automatically rolling journal (which can be configured via the ignition.conf file).
The setting on the gateway webpage only controls what is shown on that particular page.
There are a few rare use cases where an individual logger uses the logger.isTraceEnabled() or isDebugEnabled() methods to check the global state of a particular logger (which is configurable on that page in the settings menu, or via system.util.setLoggingLevel), but that's rare and not the primary action you'd undertake on that page.
Finally, if you're looking to interact with the system_logs.idb, or a diagnostics bundle, or a variety of other file formats Ignition can export that are useful for troubleshooting/diagnostics, may I not-so-humbly plug Kindling, a tool I originally created to help our internal support team that has since been publicly released and semi-officially blessed.