The fact that your photos start out as being recognised as Nikon and are recognised as Photoshop after editing suggests that the file type is being changed (indicated by a change in the three letters after the . in the file name). I would guess that the camera saves the files as JPG but that Photoshop changes the file format to its own proprietary format. Check to see if you have any options in the save as menu after you edit a photo. It will probably be possible to save as a JPG with the original EXIF data retained.
I do not know how much you edit photos but you might consider a free program such as Irfanview which provides basic editing facilities and definitely allows saving as JPG with the original EXIF data.