I found the solution. All I needed was to add the following line inside the <Product> tag in my main wxs:
<UIRef Id="WixUI_ErrorProgressText" />
Explanation
Without the above mentioned line, my MSI package was using the stock messages inside Windows Installer for ActionText, see this tutorial. It seems, that in earlier versions of Windows, these messages were identical to those supplied by Wix. That's why I got the correct "Copying new files" in Win 7 but incorrect "Copying new files File: [1], Directory: [9], Size [6]" in the latest Win 10.
After adding a reference to WixUI_ErrorProgressText, the messages defined by Wix are used (these messages are correct) and everything is OK.
Note
In Wix, there's the following template defined as well (and this has confused me at first):
"File: [1], Directory: [9], Size [6]"
But if you expect to see the file names and their sizes during installation, you're wrong. The ProgressDlg in Wix doesn't display it. If you want this info, you need to override that dialog and add the ActionData text explicitly. I didn't test it. See how it's done in PrepareDlg.
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