The reason I see, that when it is in teaser, it shows full path, like
.../sites/default/files/folder/file.odt

But when viewing the full node, the link shows system path:
.../system/files/folder/file.odt

Can somebody help to make it so, that Uploads In Teaser module would show the same path, as in the full view?

I'm using the module Private Uploads for this folder. Can it be connected?