We can sit in a chair with arms (such as a presidential throne or a Lay-Z-Boy) and we can sit on a simpler chair like a dining chair. However whichever sort of chair we are using we can sit at a table, bar counter or desk.
Under some cicumstances a person can sit on a table, desk, or bar counter but then they are sitting directly on the surface. The idea that someone could sit at a chair sounds very odd, it would imply that they were sitting on something else like a small stool or the floor.
As JeremyC says in his comment above 'sitting at the chair' may be a technical term in the particular context but it is not part of general English. I do wonder whether the legislature referred to has a history where people sat on cushions (Japanese, Arabic and Indian contexts spring to mind). In this context the idea that the president would sit 'at' a particular position would make sense. The English equivalent is "sitting at the head of the table" which relates to the position of the person relative to the rest rather than the particular chair on which they are sitting.
If this is the case then "sitting at the chair" could indicate that the historical terminology relating to sitting on cushions has been transferred to a more modern context where there is physical chair or throne.