Hi Joseph,
Yeah, I'm not crazy about the messaging system either but some of our teachers and students seem to like using it.
In this case, the students have not changed their settings for the lesson essay graded notification. The check box that you highlighted in yellow in your screenshot is not checked in my sample student but he still gets the feedback there (as well as by email).
I've tried changing it in the system so that the choosing the popup notification is disabled but with no difference in the results. The feedback message is sent to the moodle messaging regardless.
I did see a conversation about this here: https://moodle.org/mod/forum/discuss.php?d=238253 but it wasn't resolved there.
Anyway, I'm trying to get around it by having the teachers use your essay feedback block and in the lesson itself have instructions for the students to look at that block to check their feedback (and to see what they have already submitted.)
Thanks,
Janis