"Bail-in" is a newly coined term defined in the article you are reading "From bail-out to bail-in"
The article takes a long time to define the term, but it is clear that the idea is instead of giving money to the financial institution to stay afloat ("bail-out'), it would involve an emergency take-over of the institution by regulators to restructure the existing debts and write them down (declare these debts as irrecoverable).
This would immediately cause losses to some shareholders but allow the financial institution to protect the remaining shareholders rather than collapsing completely. (This is "bail-in" - staying afloat by using resources from within.)
It is a rather complicated idea, and this should probably be in the English Only forum, which is why I avoided any Italian in this response. Again, it is a word invented by the article you are reading, and it is not a normal English expression.