Choosing is not so much about whether it’s internet facing or not. From the programmer’s perspective and an administrator’s perspective there are pros and cons to both. As someone looking to self-host, if you want to run a service that works with either, I would make the choice based on what seems the most supported, or which one you feel the most comfortable looking up and performing administrative tasks on. I tend to use postgresql more just because I have more experience with it and can recommend it if that’s what you need, but mysql can be just as good or better in many circumstances. Pick whichever one looks easier to you.