I think it's strange that you can apply twice for the same award. My experience with these kind of things would say that generally, the award rules state whether or not you should apply through your current university, OR, directly to the funding agency. But not both! But maybe this award is a special case.
However, I really don't think you would be considered twice for one award so if you end up submitting two applications, there's no reason to make them different. It will be clear that you are the same person once they see transcripts, CVs etc.
But I would double check the applying thing again for sure!