One of the problems with over the boarder prescriptions is that though Canada, US, Mexico and many other countries have basically the same drugs.. We have different names.
Now a good pharmacist or doctor will know the multiple names (or be able to figure it out with some internet help) but. A US pharmacist can not dispense based on the different name. He has to have a MATCHING prescription, and, I suspect a US doctor's signature on the form.
Thus, what you did is beyond a doubt the best option, Bring paper from your "home" doctor, and visit a doctor in the country you are visiting and get new scripts to cover you while you are visiting.
But any time you cross a boarder with meds.. Mandatory they be in a properly labeled pharmacy package, Good idea to carry a spare script with your doctor's signature on it too.
Some folks, for some reason I'll never understand, insist on putting pills in unlabeled bottles or containers, Now I can understand those S-M-T-W-T-F-S boxes or the Morning/Evening boxes but transfering all the pills to one unmarked container? That is a good way to make a mistake.. A potentially fatal mistake. Plus, if you are caught with meds in an unmarked bottle, You may well be invited to spend some quality time at one of the local "Inns" (AKA JAIL)