Thanks for explaining this to me! I hate that it's public info. For my family's sake, I would want that info out there, ya know?
Basically, there are two ways that info would be made readily available to the public: if you ended up in the news/newspaper or if someone included it in your obituary.
You won't really be newsworthy unless you are somehow famous or noteworthy or you cause some sort of spectacle. If you just die quietly and there's nothing sensational about you or your method, you're not going to attract that sort of attention. If you live in a small town or something, it may get some attention, but bigger places generally only care if other people are involved, it's murder, or you kill yourself in a really bizarre way. An easy way to judge is just pay attention to that kind of thing and you'll see what gets reported.
All you can really do is request that nobody do the obituary (or say what you want in it) and hope they respect that or write one yourself and hope they use it. Obituaries tend to gloss over the specifics and say things like so and so died suddenly or unexpectedly or something like that. Your family isn't going to put anything in it that they don't want other people to know about.