The box is sometimes visible for a friend, as snapshot #1 & #2, and sometimes not visible for another friend, as snapshot #3. I don't know why.
Why is that?
"Search your friend" box is visible
Not visible for some facebook friends
This means your friend in snapshot #3 has prevented you from seeing his/her friend list.
Maybe it is set to
I'm sorry mate but I think I wasn't able to grasp and fully understood your question.
But just my experience in using fb as I am not really active on being there, I am only using the FB official search bar at the top of the page and type "find friend" and it will give you the page where you would be able to find some of your other friends through your SNS or mail contacts.
Goodluck FB'ing! :)