AA:
It looks to me that the tree did split but that it wasn't a barber chair.
There appears to be a twisting split at about 4:25 in the video that comes down from the upper left. This split does not remain on the stump but is taken off with the fall.
You can see it by its light color.
A barber chair is a split where the fulcrum is moved up the tree.
In this case it was the back strap sponsoring the split, as it looks like the tree was pulling away from the strap, but it did not change the hinge. Another way of thinking of this is that it didn't create a method for the tree butt to either be suspended in the air or slam down on the faller. So it doesn't appear to be anywhere as dangerous as a 'chair but I don't like it. That doesn't mean I could have done better, just that it could have unintended consequences that I'm not familiar with.
I can't say but there may have been a way to prevent or further reduce this splitting from occurring. Perhaps to continue cutting the back strap till it popped cleanly? Then again, getting away from the stump and not getting the bar grabbed by the tree have merit.
It looked like leaving the strap at that corner was appropriate with what we saw of at least one lean on the tree, I would have suggested making all of the back cuts, both plunge (bore) and strap cuts from the side opposite the lean. (By using the sights while completing the face cut then matching that with the plunge/bore cut one does not need to even walk around the tree to check the off-side holding wood.) Avoiding cutting under the lean is desirable not just from bar pinching but from limbs and tops falling on the cutter. There is a tendency in the plunge cutting world to almost brag about using shorter bars. A longer bar means not just distance from danger - while bucking for instance - but also not placing yourself in danger.
Long Bar = safety.
One error that I would ask you to consider is that you didn't look up even once while making any of the back cuts. Your vertical awareness moments were only after completing the strap cut. Your plunge cuts were made at an uncomfortable height that required more concentration. That sort of multiplied the distraction and that height, closer to your eye level, can cause even more distraction with saw chips/dust being blown into your face.
In any case I would suggest the plunge cutting was highly desirable on this tree.
Just a thought for the reader, if any complication in a tree/snag being dropped takes away from your vertical situational awareness to the point you are oblivious of what is happening above while cutting, that might be a good tree to step away from.