Folder from external drive into Public Library

open your library and right click on videos/movies folder and select properties there are options to add a folder to the library , it doesn't really add the folder but watches the contents, you can put folders from your external drives also
also be careful deleting in the library as it deletes from the drive also, at least that happened in my case i ended up using easeus recovery

The problem is that you aren't using Libraries correctly. Libraries are only picture icons that represent the actual files that reside in your User folders.

Libraries aren't made for the stardard drag and drop file movement and there are good reasons for this. You would be far better served to use the User folders My Document, My Pictures, etc. to move those files around using the drag and drop method.

You would also be better off not mixing your Public folder with your Libraries and visa versa, it's best to keep those seperate because they both work off of a different set of sharing rules. Libraries aren't made to be used for public sharing, they are only made to be used for the LAN computers which are connected to a common Homegroup using a Homegroup password. This adds an extra layer of security to the Homegroup.

You can move files inside of Libraries but you need to do it by using the tool bar on the upper left pane, it's not enabled by default. The drag and drop method does not work correctly with Libraries, it's made this way for a reason.

In case anyone wants to know how to send or copy files to and from the Homegroup Libraries there is an easy way. You need to use the menu pane for this.

One problem is that the Menu pane doesn't show up by default but thats easy enough to change. You can also go to folder options then view tab and select the "always show menus" option but that isn't checked by default either.

I'll go over the steps needed to access the copy to and move to folder options just so it's very clear for anyone who reads this.

With a Library open as in Library>Documents look to the left and click on Organize> scroll down to > Layout then click to make sure the Menu pane is showing. Now that you have the Menu pane checked you will see file,edit,view,tools,help at the top of the page on the left.

Next highlight the Library folder or file you need to move or copy, then click on the edit button in the Menu pane and choose copy to or move to folder option, click on Homegroup then choose the persons name, then choose the Homegroup folder you want to sent it too and hit the copy button at the bottom of the page. You can easily move files around between Libraries using this method.

This tutorial from Brink also shows you how.
Context Menu - Add Copy To Folder and Move To Folder