I was offered this puzzle verbally by a Captain I was flying with on a trip (I'm a copilot, it was a 2 man crew on a domestic trip). I was told to find the solution in three moves, so never deliberately sought a 4 move solution. And I had to do it without any props, as it was a verbal challenge.
I set out with crazy diagrams that looked nothing like yours so I'm not sure if my solution is entirely different or not! I can say, my mindset was totally different. It quickly dawned on my that I had to garner AS MUCH AS POSSIBLE per weighing. That meant finding a grouping that eliminated as many pieces as possible per move. Grouping them into three groups meant an equal weighing eliminated eight, or an unequal weighing eliminated four.
That spawned two new challenges. Finding and determining in two moves, the odd piece from four. Or, find and determine in two, the odd piece from eight...
The first was easy, the second then starts to look like your reasoning... I need to find my notes and try to diagram them digitally to be sure, it's been a while, I just found this site!
A fantastic puzzle, I intend to build one to put on my coffee table.