When they say "the original container" they are referring to a container of some sort with the prescribing information.
I take a few meds and some are fairly large. Because I travel for up to three months at a time, I have to take quite a bit of meds. To save space some go in a ziploc bag. My pharmacy prints out an extra set of labels for each drug and I put them all on one piece of paper. That is my proof of what each drug is and that I am the person who it was prescribed for. What is the difference between a bottle and a ziploc as long as I have the pharmacy label?
Last year I had to take a couple of prescription lotions with me. They went in their own small ziploc bag with the cut out prescriptions from the box each lotion came in. I didn't take the entire box. TSA didn't have a problem with them nor did any European airport security.
Non are opiods so I am not too concerned. Some countries, Japan is one, are very strict even with prescription meds so one has to be careful. But Japan, last I checked, was not in Europe.
Now, if you just took the pills and put them in a ziploc bag without any prescription info, then you might get in trouble. Just have the label and you will be okay.
By the way, most physicians have a federal license to prescribe medication so if your doctor writes a prescription it can usually be filled in another state if necessary. In fact, if you are traveling around the country (USA) and need your meds, just go to a local branch of your pharmacy, if it's a national chain, and they can get you a refill.