I turn off every audible notification on my phone that is not one of the following:
1) another human trying to communicate with me in real time (phone call, text message, or something equivalent)
2) an alarm I previously set
There is nothing I find more annoying than my phone beeping or buzzing over something that does not require my immediate attention. The last thing in the world I want is to be woken up by my phone because I received an e-mail letting me know a game on my Steam wishlist went on sale.
That's the rub -- for your #1, I largely agree. Someone reaching out to me via instant messenger is a notification I don't mind -- it's real time. But when I'm actively using the software on another system, namely a computer or game system, the notification serves no purpose.
I definitely need to go in and reduce some settings that I haven't dealt with due to laziness. And there are some apps I've had to essentially remove because they're more interested in annoying me than being useful.
There was an app called NextDoor that is supposed to be a sounding board for your neighborhood. Things like "hey, I lost a dog"...."there was a break-in last night', etc. Firstly, it has information that goes way, way, way beyond what I'd call my neighborhood -- things from cities away. And it kept buzzing with notes like "Hey, is anyone else from North Jersey? I just moved here." In my opinion, the general public doesn't really understand social media and what outlets are appropriate for different kinds of messages and conversations.
Super annoying.
And Facebook is the G-D worst, which sucks, because I think it can be such a useful tool. My "feed" consists of people who I loosely know commenting on subjects that they subscribe to and I don't. So, let's say I'm a fan of the Dolphins. It's fine to push me discussions within that topic. But along with that, I get a bunch of "things you might like" posts from some chick I went to high school with commenting on a meme from "Wine Moms" or some stupid shit. A lot of it is "only geniusiz (sic) can name a word that ends with 'ant'" and then 2000 stupid-ass responses." It's awful. I would just bail on it, but I need it for some clubs I'm in and I do like the idea of having some form of loose communication with pretty much everyone I've ever met, just as a way to reach people, learn about major life events, etc.
I also find that some apps have too black/white a line for notifications. This is a hypothetical, but let's say an app like "Offer Up". I wouldn't mind if someone posts a question or makes an offer on something I'm selling -- that's valuable information to me, so I would like to be notified in real-time. Yet, I definitely don't want a notice that says "new Aquarium related items for you" buzzing in my pocket because I looked for a fish-tank 6 months ago.
In most of these cases, I bet there are options to get the app to work smarter for you, but I don't often take the time to delve into anything other than the default settings.