Hi. This forum is for beginners trying to learn the basics of Photoshop. I'm moving your question to the Photoshop General Discussion forum for specialized attention to your situation.
It sounds as though Windows is still showing the PSD files as being associated with PSE 7.0. Have you gone to Windows Folder Options, and reset the File Type Association to PS CS6?
when I look for or open a PSD file from CS6 that my PSD files are defined in the "Type" column as follows "PhotoshopElements.PSDFile.7"
Am I correct what your referring to in the above statement is the description for PSD file types in your window's file explorer window details view. If so that just a string that was added to the windows registry for the file explored and not removed and is now the default description for file type PSD. The may have happened when you removed CS5 and it restored the previous description for it though the current one was Adobe Photoshop Image file 12. and it replaced it with the one it hade replaced in your case "PhotoshopElements.PSDFile.7"
I have had a lot of problems removing old version of Photoshop using the normal uninstall process. Problems were always related to Window registry updates made during the uninstall process. These days I leave the registry out of it for the most part. I remove the uninstall entry in the registry for the versions I want to remove then just wipe the Photoshop version install tree. I leave common file and registry intact. I also left CS5 installed for there are just too many bugs in CS6.
Most likely if you just reinstall CS6 on top of itself you will loose nothing and the registry will be updated with CS6 description for PSD files which is "Adobe Photoshop Image 13"... You do no need to its just a description.