As commented in
MDL-63303, the online/offline distinction has been removed on purpose from the messaging user settings page, but it is still working in some cases, like when the user hasn't any default messages preferences and they are defined with different values in 'Site administration | Plugins | Message outputs | Default message outputs'.
This confuses a little bit because, by default, on a fresh installation, users will only receive email notifications for personal messages when they won't be connected. However, if they disable and enable 'Email' from the new messaging settings page, they will receive email notifications for private messages always (regardless if they are online or offline).
We should review this part to make it more consistent from the user's POV. For instance, we could display the online/offline information (when it's not the same) to let know it to the user, add some help text to explain how it works or even add it again to the settings page or remove it from the 'Default message outputs'.