Facebook – What Does “Turn On Notifications” mean on someone else’s Facebook post

By default you will see anything from that person that is publicly visible, including public comments. To turn these off, go to their profile or timeline and find the "Subscribed" button at the top, or hover over their name in one of their posts. While hovering over the "Subscribed" button, go down to "Comments and Likes" and uncheck it to stop seeing their Comments and Likes in your News Feed.

It looks like Facebook has done another update so everyone gets more notifications. This thread on the Facebook forum has a couple of solutions suggested, this one seems to have the most love:

Go to the top of your screen and click the little globe symobol where you can view your recent notifications. Hover over one of the them on the top left and click the X. It'll ask if you want to turn off "top friend" activity notifications. Click yes. They will still appear in your news feed, you just won't get notifications for it all.

On a side note I think the icons you've circled in the first picture have more to do with identifying the notification than close friends (photos & status get the star, links get the pin).

Updated

Click the Notifications globe at the top of the Facebook window and when the drop-down drops click the Settings link (the little X's next to each notification have been removed recently). In the "What You Get Notified About" section, you can edit and turn off "Close Friends activity"