Oh, FFS, if you want to help, then stick unimportant mail in a "maybe when I get around to it" queue. Don't ask/train people to be obnoxious.
The second type is people who receive a lot of correspondence – people expecting a response, or otherwise need action.
The people who claim that it is the receiver's responsibility to process all of their email almost exclusively fall in to the first type, because nobody who falls in to the second type can reasonably process it all.
Surely something like a contact form could put any correspondence in some kind of queue that a person could just get to as they have time, instead of just using email directly. Or having an email account just for correspondence with the public so it isn't mixed with other types of email. Then just time block some time to reply to people each week.
Then hire a secretary to manage the inflow of data sent to this person, or maybe change the org structure so that the individual needs less direct correspondence. Frankly people are trying to fix systemic not-enough-time-single-point-of-failure org structures with process Band-Aids that only lower the net productivity of everyone involved.
UNLESS dealing with email is your primary job responsibility. Then I would fully expect you to have your act together.
I do think it's your responsibility not to encourage people to spam everybody else, even if getting repeated requests so you can "help people when you're not busy" tickles your ego. If you don't want to help them with whatever it is you do, let them go somewhere else.
You are also making yourself busier, which kind of makes a person question your sanity if you really are busy already. Even ignoring a message is an effort.
Do enough people not have conversation threading that this is an issue? One unread message takes up no more room in my inbox than three unread messages in one conversation.