If you take a medication that is a controlled substance you should always keep it with the vial. Some medications are light sensitive and dark vials protect from this. I agree that a copy of the script is important. Some countries are strict about this. I would not venture into Japan, Singapore the Middle East or Asian countries without the original vials. I was planning a trip to the world master games in Japan and they were very clear about original vials. At minimum, in an emergency situation having clear info about what medications are in use can help a doctor or EMT with triage.
If you do opt to pre load your daily pill container, make sure to store it in a controlled light situation, like a drawer or inside a travel pouch. As kindly suggested above, having an extra printed label from your pharmacy is prudent