Changing DB to local + backing up

Hello,
I just noticed that only two of the 10 MySQL databases configured on my server are listed as not-remote-accessible and only these have the name of the server in the corresponding column "Server". Coincidentally, only these two are backed-up by the automatic feature available in the Backup page of a website.
When I open the configuration of a remote-accessible DB, the name of the server IS there; but try as I may, I cannot get it not to be remotely accessible - i.e. even if I unclick the "Remote Access" box AND clear the "Remote Access IPs" field, when I click "Save" nothing is changed.
So here are my questions:
- is there a real connection between a DB being remotely accessible and it not being backed-up?
- how do I get all DBs to be local so that it can be backed-up?

Thank you!

 

All DBs have both a website AND a server selected in their settings - or at least that's how it appears in the config page. In the overview table, the website columns are also filled with the corresponding website. However, only two DBs have a server name in that column.
Any suggestions?

EDIT:
Here are two pictures showing the overview table and a single DB's config page:

 

I've retried to alter the settings, but nothing seems to do the trick. I also noticed that phpMyAdmin dsiplays some errors, could it be that they are the reason for that?

Code:

$cfg['Servers'][$i]['pmadb'] ...    not OK [ Documentation ]
$cfg['Servers'][$i]['relation'] ...    not OK [ Documentation ]
General relation features: Disabled
$cfg['Servers'][$i]['table_info'] ...    not OK [ Documentation ]
Display Features: Disabled
$cfg['Servers'][$i]['table_coords'] ...    not OK [ Documentation ]
$cfg['Servers'][$i]['pdf_pages'] ...    not OK [ Documentation ]
Creation of PDFs: Disabled
$cfg['Servers'][$i]['column_info'] ...    not OK [ Documentation ]
Displaying Column Comments: Disabled
Browser transformation: Disabled
$cfg['Servers'][$i]['bookmarktable'] ...    not OK [ Documentation ]
Bookmarked SQL query: Disabled
$cfg['Servers'][$i]['history'] ...    not OK [ Documentation ]
SQL history: Disabled
$cfg['Servers'][$i]['designer_coords'] ...    not OK [ Documentation ]
Designer: Disabled
$cfg['Servers'][$i]['recent'] ...    not OK [ Documentation ]
Persistent recently used tables: Disabled
$cfg['Servers'][$i]['table_uiprefs'] ...    not OK [ Documentation ]
Persistent tables' UI preferences: Disabled
$cfg['Servers'][$i]['tracking'] ...    not OK [ Documentation ]
Tracking: Disabled
$cfg['Servers'][$i]['userconfig'] ...    not OK [ Documentation ]
User preferences: Disabled

Any help would be appreciate it.
Thank you!