Subquestion to my question [1]:
All definitions of (MS SQL Server) index (that I could find) are ambiguous and all explanations, based on it, narrate something us
The definition of an index is the first part of Wikipedia definition "A database index is a data structure that improves the speed of data retrieval operations on a database table at the cost of slower writes and increased storage space."
Then you have unique indexes, as a special kind of index, which ensure that indexed values are unique.
How it's implemented... depends on the DBMS. But it does not change the definition of index, or unique index.
As an implementation detail, MS SQL allows non-clustered (the usual kind, which is a tree with pointers to the actual row contents in a separate space, which you numbered 2.), and clustered (where rows are stored in the index, according the indexed value, which you numbered 1.) indexes.
So an non-unique non-clustered index is just (conceptualy) a tree of values with, for each value, a set of pointers to table rows containing this value.
An index is a data structure designed to optimize querying large data sets. As such, no claim is made about whether or not anything is unique at this point.
You can definitely have non-unique non-clustered indices - how else could you index on lastname, firstname ?? That's never going to be unique (e.g. on Facebook.....)
You can define an index as being unique - this just adds the extra check to it that no duplicate values are allowed. If you would make your index on (lastname, firstname) UNIQUE, then the second Brad Pitt to sign up on your site couldn't do so, since that unique index would reject his data.
One exception is the primary key on any given table. The primary key is the logical identifier used to uniquely and precisely identify each single row in your database. As such, it must be unique over all rows and cannot contain any NULL values.
The clustered index in SQL Server is special in that they do contain the actual data in their leaf nodes. There's no restriction up to this point - however: the clustered index is also being used to uniquely locate (physically locate) the data in your database, and thus, the clustered index must be unique - it must be able to tell Brad Pitt #1 and Brad Pitt #2 apart. If you don't take care and provide a unique set of columns to your clustered index, SQL Server will add a "uniquefier" (a 4-byte INT) to those rows that aren't unique, e.g. you'd get BradPitt001 and BradPitt002 (or something like that).
The clustered index is used as the "pointer" to the actual data row in your SQL Server table, so it's included in every single non-clustered index, too. So your non-clustered, non-unique index on (lastname, firstname) would not only contain these two fields, but in reality, it also contains the clustered key on that table - that's why it's important the clustered key on a SQL Server table is small, stable, and unique - typically an INT.
So your non-clustered index on (lastname, firstname) will really have (lastname, firstname, personID) and will have entries like (Pitt, Brad, 10176)
, (Pitt, Brad, 17665)
and so forth. When you search for "Brad Pitt" in your non-clustered index, SQL Server will now find these two entries, and for both, it has the "physical pointer" to where to find the rest of the data for those two guys, so if you ask for more than just the first- and last name, SQL Server could now go grab the whole row for each of the two Brad Pitt entries and provide you with the data the query requires.