I did some checking after making the video, and my confusion over "tetrominoes" versus "polyominoes" can be explained simply by how many squares each shape contains. Tetrominoes are the shapes used in _Tetris_ , and they are tetrominoes because they are made of 4 squares each, "tetra-" being an English prefix derived from Ancient Greek meaning "four". Pentominoes, likewise, are similar shapes containing 5 squares each. "Polyominoes" is a more inclusive word which refers to such shapes regardless of how many squares they contain. Since the number of squares which comprise the shapes in this game's puzzle are variable, it thus follows that they are polyominoes.
Also, apparently both "Tower of Hanoi" and "Towers of Hanoi" are correct, but the former, as used in this game, seems to be more common. I guess I grew up in a family of hipsters.
At present, LateBlt has 3,569 views spread across 8 videos for The Island of Dr. Brain, with the game making up 4 hours of published video on his channel. This makes up 1.27% of The Island of Dr. Brain content that LateBlt has uploaded to YouTube.