It's not the media that makes the difference it's what you play it back on.
The Cinavia signal will remain present unless it's removed and in fact it's particularly robust even surviving all sorts of conversions/transformations.
It's your player that detects and reacts to the Cinavia watermark so if you play it back on a device (or software player) that supports Cinavia then you'll get the warnings.
By way of example, if you convert a Blu-ray movie to DVD and play it in a stand alone DVD player Cinavia won't be triggered as DVD players aren't capable of supporting Cinavia. If you take the same DVD though and play it in a Blu-ray player that supports Cinavia then it'll be triggered.