We're also talking about humoral immunity, which relies on antibodies. There is no self recognition involved there. As for cell-mediated immunity, it's true that the self MHC is part of what's required for recognition, but MHC alone is not sufficient - you need the foreign antigen loaded into the self MHC for recognition.

As for MHC and HLA, they are not two different molecules. MHCs are HLAs.