Should I run brew or update java via web or some other method?
You can update Java by downloading the newer version from Oracle directly When you install the installer would delete the older version for you.
Or you Mac has built in system that would fetch and update java from Oracle for you for you. Go to System Preference > Java > Update > Update Now. It would also be a good idea to turn on "Check for Updates Automatically" that way when a newer version of Java comes out your Mac would ask you to install it and it becomes a simple one click install.