The label on a prescription drug in the United States IS a copy of the prescription. It has the same components: drug, strength, sig (directions), quantity, patient, and prescriber. There is no reason to have any additional paperwork as long as this is all legible and intact.

Pill organizers are immensely useful, but the just can't be used for international travel.

As for other countries:
Australia is exceptionally strict. One has to declare everything. But as long as it is declared and legal in Australia, it is fine. But you must declare it.
The UAE, among a few other countries, have strict rules about commonly-prescribed drugs in the US. This is especially true for amphetamines used to treat ADHD. I have provided written letters to patients, and that has been adequate.