Proton Mail is still doing it right. Go look at what they're doing, learn from them, and then also do it yourself.

When I sign in, and I un-tick the "Keep me signed in" tick-box, I do not want to remain signed in -- not even halfway -- when I sign out again.

Which part of the difference between "in" and "out" do you not understand? Go watch Sesame Street, they've got some good episodes on the difference between "in" and "out". ...or Tellytubbies... or one of those educational programs for babies.