Having watched an uncompressed version of the video, I see what people are seeing now - was literally 'blink and you miss it'. I initially thought that people were saying the shot of him before he heads up the stairs was Arthas (that's the shot the link shortcuts to) and I sat here like... how the hell does anyone think that is Arthas.
If you look at the hair when he's going up the stairs, it is Anduin's hair. Arthas' is slick and all one length where as Anduin has... bangs/layers? that are still visible as he goes up the stairs. But for some reason he is missing the brown cloak that he has before heading up, and is still wearing at the top.
That's definitely a production error. Its like they switched to the standard version of his model, then switched back to the cloak at the top. I'm honestly surprised that got past Terran Gregory & crew.