There are two ways Reader can attach documents to an enabled PDF, either as normal file attachments or as comments. The latter can be enabled with Acrobat or LiveCycle Reader Extensions, but the former can only be enabled with LiveCycle Reader Extensions. It is not done through the security settings in any case.
OK, that's the reason for the difference you're seeing. You can enable a document for commenting using Acrobat 9 Standard and Reader 10 will be able to add attachments as comments. To do so in Reader, select "Comment > Annotations" and you'll see the "Attach File" tool. It looks like a paperclip.
There is no Reader Standard. There is Adobe Reader and there is Adobe Acrobat Standard and Adobe Acrobat Professional. Adobe Reader is a free program, Adobe Acrobat is a commercial product used for creating pdfs and adding enhancements to alreadly created pdf files.