Data Source path in Pivot Table changes to absolute on its own

Hello.

I have a .XLSX file, that was created long time ago (I don't even know in which Office version, but definitely not 2013), and maybe even was a .XLS file at first.
So it's a 4 MB file with 16 Sheets and 8 Pivot Tables.
All of the Pivot Tables use other sheets from the same file as Data Source.

Data Source for some of them look like this: 'Sheet3'!$A:$E

Everything is fine when I save the file, and open it from saved file.
But as soon as I try to move the file elsewhere, or rename it, or email it - all Data Source paths change to something like this: '\Users\Sergii_Litnevskyi\Desktop\New folder\[FileName.xlsx]Sheet3'!$A:$E
And it happens with all Pivot Tables. The problem is that it links to an old file path, where the file does not exist anymore. And it links to an external file, which is not what I want.
If I Save As and select different path and filename - then it works fine. So it's a workaround for renaming and moving files, but not for sending them to other persons.

I've read some forums, and people recommend disabling "Save external link values", but it does not help. It is already turned off in my office, but it keeps acting weird.
So what I need is: Save the file, close it, rename it, move it to other place, send it over email as attachment. And then I want to have the same Data Source path in my PivotTables as I had before I saved the file. How can I do it?

My Office version: Microsoft Excel 2013 (15.0.4454.1503) MSO (15.0.4517.1005) 32-bit

I can even record a short video to show what happens.
Actually, I just did it. You can see the video here: http://screencast.com/t/qMBild3ck9b
It is rather big - 23.8 MB.

Let me explain what I showed there:
I opened my original file. I showed that there are Pivot Tables, whose Data Sources are in the same file, on various other sheets.
I showed this for all of the Pivot Tables in the document.

I saved the file using Save As in a different folder and under a different name (TEST.xlsx).
I then opened that saved file to show you that it is fine, and the Data Source path for one of the Pivot Tables is the same as it was in original file. It is the same for all of the other Pivot Tables.

Then I closed, and simply renamed the file to TEST123.xlsx.
Opened it, and first thing wrong - Security warning.
Then I got ‘Cannot open PivotTable source file ….’ messages. And, as I showed, now all Data Source paths have been changed to full paths of the file, that was created by Save As (TEST.xlsx) from original file.