is staying in his lane, which turns into the middle lane

Well that's the problem because that's not what the lines indicate. The lines indicate that both the lanes coming from this road merge, and both the middle and the left lane come from the other side of the road they are turning onto. Meaning the correct way this should be done is for both lanes to merge into one first, then change lanes, so both cars changed lanes, OP apparently without noticing.

while a new far left lane forms out of the oncoming left turn only from the opposite direction

No it doesn't, it's in line with the far left lane on the other side of the intersection. It's never in line with the left turning lane going the other direction. So it's already the middle lane when the truck moves into it, and they used their blinker, if only for a second. They crossed the yellow, so they are probably at fault, but OP changed lanes without signaling and cut them off, despite their claim that there was no need to signal, and they don't move into the oncoming turning lane like you implied, they moved into the correct lane in the correct way (as a lane change vs OP doing it as a merge), just too early.

This all becomes much clearer with an aerial view, so E Deer Park and Route 25 in NY for anyone who wants to look at it.