You are heading in the right direction but see this:
The driver got badly hurt. The driver is both unique and specific - there can only be the one driver.
The/a consulting manager. We could use both initially because we have no knowledge of the managerial set up. Saying the manager is used loosely by those who have little or no knowledge of the organisation. However, if you know that Mr X is one of a number of managers then you would use his name instead of the/a xxxxxxxxxx manager.
The houses on the hill (assuming that the listener knows which hill you are referring to) are both unique and specific, unless you intend to refer to some of a number. Then you might say those 5 houses right at the top of the hill....... Quite specific and unique in that whilst other house may look the same they are not actually the same.
"If all of us are five going to the same school and say "We are the students of ** school", does that mean we are "unique" or "specific" students of the school?"
Firstly, do you mean that all of you are 5 years old, or, that all 5 of you are going to school? You are referring to 5 specific persons each of whom is unique.
Student or Learner