What to look for when purchasing Thick Thread
One thicker variety of embroidery thread is called perle cotton. That might be what's in the video, though I can't quite tell. It is sized by numbers, where a smaller number indicates a thicker thread. You might be interested in a size 3 or 5 perle cotton.

If what you got originally was 6-ply embroidery floss, you wouldn't see sizes listed because it only comes in one size. That material can also be separated so you can use fewer strands, which is not true of perle cotton; that seems to me to help perle cotton threads stay round when you stitch with them, instead of flattening out.