As noted in the comments, more pictures of the piano or more information about the maker and model might be helpful.
Should I unscrew the thing connecting the fallboard to the body?
You can clearly see gouges on the screws from their having been removed and replaced with an ill fitting screwdriver, so that is apparently a solution others have chosen. A professional tuner would likely do it; removing screws and parts of the case is frequently necessary for tuning and always necessary for doing more involved work such as adjusting the action.
If so, is it better to unscrew it from the body, or from the fallboard?
I would do whatever is easiest, especially since I would want to minimize the potential for further damage to the screws or the holes in which they sit.
(I would also consider folding the front panel and fallboard down without detaching them from the side board, but I suspect that I would not consider it for very long: there's a good chance that it would increase the risk of damage rather than reduce it, and it is also likely, depending on the size of the instrument, to make tuning more difficult rather than less as it will make it difficult to move from the keyboard to the strings.)
I would also be careful to find a screwdriver that fills the slot in the screw head. It's better to have the right tool for the job, and if you're planning on maintaining the tuning yourself you may find yourself doing this several times a year.