It's worth noting that in many cases, if not most, we simply don't know the exact answer to such excellent questions as the one you ask.
Note that the book you mention (ISBN-13: 978-0471409762) was written around 2000 and appeared in 2002: that's a really long time ago in terms of all the amazing new instruments which have become available.
For example, while the Hubble telescope was operational when that book was written, the amazing GAIA telescope (link), which would perhaps be relevant to your question, only started operating around 2014.
The youtube video you mention seems to be more up to date - a couple years ago. But then, the author (even if he's an expert) may not especially be keeping up with the latest research and thinking on the particular star system you mention.
So it's worth bearing in mind that many of these questions, we just don't know - it's something that is being actively studied now!
An interesting point to consider: we're not even really sure of the distance to the Polaris system - for goodness sake! That may be a surprise. It looks like estimates of the distance - and they are just estimates - range from roughly 350 to 450 light years. We could be out by a hundred light years.
Regarding your specific question: how many stars are in the system. Don't forget when you look at a specific star, X, let's say it is 500 ly away. In the same "shot" you see a vast number of stars right next to star X. Some of those may be "next door to us" and some may be tremendously further away. It is by no means a "sure thing" to determine if two stars which appear right next to each other, are actually anywhere near each other.
(A good example of that is ... you know the Andromeda Galaxy, which you can even see with your eye. At first, astronomers had no clue if it was a thing (sort of like a star, but cloudy), and as close to us as the other stars, or, if it was an enormous object an incredible distance away. Obviously we now know it is truly huge thing an unbelievably long way away ... but, we only learned that in about 1920 - !!! Not even 100 years ago.)
For a short point on your specific question: note that the book by Plait is some 15+ years old, and the video talk is more recent; in the first instance you can go with the info in the video talk. It's a good example of how astronomy knowledge is dramatically changing all the time.