From what I can tell, I have the “ideas” and “Users” categories set to “Normal”, so I should not be getting any notifications except when someone mentions me or replies to me, but I’m still getting a “New” notification on every new topic in these categories.
Am I misunderstanding what “Normal” means? Am I getting the settings wrong? I don’t want to have to mute the category, because it won’t show up in the UI at all, but I also don’t want notifications.
“Normal” is the default setting for any given topic (a top-level discussion, like this one we are in now started by your question). It’s not something that applies to categories and it won’t hide the topics or not show that they have been active under the Unread or Latest groupings.
What you want to do is only possible by setting those categories to be muted. That does mean the categories won’t show up on the “Latest” and “Categories” pages, but you can still browse them via the preferences -> categories -> muted topics.
That doesn’t really make sense based on the options given. If “Normal” is the default setting for every topic, what is the difference between “Normal” and “Watching First Post”?
Ugh. The use of “notification” rather than “e-mail” is very misleading, particularly for those of us who have all e-mail notifications turned off.
It is very unfortunate that there is no option to turn off notifications other than hiding the entire topic. It’s very strange to me that they had the foresight to include the distinction between “Watching First Post” and “Normal” in the UI, but not to have a way to have it apply to notifications on the site itself.
Are you referring a different thing as notification? IIUC Discourse notification is the bubble overlay number appearing on the top right profile icon (you said you turned notifications off, so maybe you never saw that). The number after each category shows the unread count (not sure if this is the correct term), and I believe Discourse is designed around the assumption that the user don’t need to care about them.
Huh? You don’t need to care about what things you haven’t read? That sounds strange (to put it mildly)…
I chose a bad expression 
To me, the unread count is unlike the unread count of one’s email inbox, but more like maybe if my Twitter timeline has a counter. I organise my email similarly; I create filters for mailing lists I join, and make emails skip inbox and go directly under their respective tags, unless they specifically include me in the To field (which makes them show up in inbox). I skim through them when I have time, but only pay (semi-)immediate attention to the inbox’s unread count.
I make sense of how Discourse work based on that model: the unread count indicates how much I need to catch up, but I put immediate attention only to notifications (blue bubble on my head).
Yeah, I think that’s probably pretty fundamental to my frustration with Discourse. My workflow is to keep my unread count at zero at all times, but what I’d like to do (and email lets me, but Discourse makes it hard) is to focus on a per-category basis (so if I have little time, maybe I’ll read all the unread messages from the packaging category, but ignore the Discourse Feedback category). If I run out of time, I’d like to mark what I was part way through as unread, so it reappears in my “todo” list for later - but Discourse has no “mark as unread”.
There may be a way of doing something similar in Discourse, but the mental model it’s based on is so different from what I’m trying to achieve (driving my process by “what is unread” seems very foreign to Discourse) that I’ve never found a particularly efficient and scalable approach 
Perhaps you want to use Discourse as a mailing list instead then? The Mozilla community has written up a FAQ on how to use Discourse via email that may be of interest.
Nope, I tried that, it’s the worst of both worlds. And that FAQ in particular seems to rely on interacting via the website as well as via email, which is definitely not what I want.
I don’t seem to be making myself clear - it’s not so much the mechanics of the individual tool that’s a bad match for me, it’s the underlying model of how conversations, quoting, keeping up with discussions, etc. work.