I carefully watched your video and then browsed the site as well. What I noticed is that the K2 sub-categories without issues are not linked to a menu item. So my guess is that because the site's structure is complex/large, it's most likely that the "bad" K2 links/categories are simply referenced by old menu items which our outdated. Which explains why the items loaded are correct but the titles in the URL and breadcrumbs is wrong. The last 2 are properties of the Joomla menu system (not K2) and this is evident by the short URL structure these have.

E.g. /beads-and-jewelry.html (bad title/URL, correct fetched items) vs /videos/painting-on-glass-videos/itemlist/category/66-painting-on-glass.html (good).