Let us distinguish. Up top where your avatar/username shows to the right are a little inbox icon and a bell icon. The former are your messages (private or direct messages /pms / dms) and the bell represents notifications. Notifications are for various things that happen on the site mostly related to interactions with other members. So a new message could be one reason you get a notification, a connection request could be another, a connection accepted, a new follower…
If either icon has a little bubble with a number attached to it, that means there are that many “unread” and clicking that icon will show you the unread messages (inbox icon) notifications (bell icon)
As far as I know, there is no place where a little eyeball thingee appears for messages. There is however such an icon for notifications. When you see the eye icon it means that a notification is unread. if you click it you’ll see an eye with a slash through it, meaning that it was already read. If you click that, it will transform again back into the icon without the slash (unread)
So it could be that you marked some notifications as read, and these notifications were for messages that you received. The actual messages would remain unread until you actually read them. In the messages inbox, there is no eye icon. But you can mark individual messages as read/unread. If a messages is unread and you click the message, it will automatically go to read status.