I don't think physicist really misinterpret this if you asked them. The problem is, as you say, that the "past" and "future" only have a physical meaning inside your own light cone. As alluded to in the video, you cannot compare space and time differences without defining a common reference frame, so it doesn't really make sense to talk about a past and future in spacelike separated regions of spacetime.

Special relativity is great because it produces some very non-intuitive conclusions about our Universe, and it's always lovely to explore new mind-bending questions arising from these conclusions, especially in outreach pieces like this. I think conceptual discussions like these can be very enlightening, but its not something that keep a lot of physicists up at night.