Can I define the permission of a folder that was shared with me?

Only if the owner of the folder allowed you to do so. The owner of the folder needs to tick off all of the boxes in the category Permission, to grant you the authority to do so. If this is the case you will see it if you click on Share next to the shared folder and click on View people. Under Permissions, it will say extended next to your details. The default permission is modify and it will not allow you to define permissions of a folder.