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What are Covering Indexes and Covered Queries in SQL Server?
Only the third example is covered. To be covered, a query must be fully satisfied from the index. Your first example produces results that are entirely within the index, but it needs information that is not part of the index to complete, and so is not covered. To match your first example, you need an index that lists Col3 first.
One important feature of indexes is the ability to include a set of column in the index without actually indexing those columns. So an index example for your table might look like this:
CREATE INDEX [ix_MyTable] ON [MyTable] ( [Col1] ASC, [Col2] ASC ) INCLUDE ( [Col3])
Now samples 2 and 3 are both covered. Sample 1 is still not covered, because the index is still not useful for the WHERE clause.
Why INCLUDE Col3, rather than just listing it with the others? It's important to remember that as you add indexes or make them more complex, operations that change data using those indexes will require more and more work, because each change will also require updating the indexes. If you include a column in an index, without actually indexing it, an update to that column still needs to go back and update the index as well, so that the data in the index is accurate... but it doesn't also need to re-order the index based on the new value. So this saves some work for our database server. To put it another way, if a column will only be in the select list, and not in the where clause, you might get a small performance benefit by including it in an index to get the benefit of covering a query from the index, without actually indexing on the column.
It is not just the where clause and select clause. A group by clause also needs its columns to be covered by the index for it to be a covering index. Basically, to be a covering index, it needs to contain all the column used in the query for a given table. However, if you don't include them in the right order, the index won't be used.
If the column order in the index is (col1, col2, col3), then the index can't be used for query one since you are selecting by col3. Think of it like a phone book sorted by last name, then first name, then middle initial. Finding everyone with a last name Smith is easy, finding everyone with the first name John isn't helped by the sorting, you have to read the whole phone book. Same for the index. Finding a col1 value is easy. Finding a col1 value and then col2 values is fine. Just finding col3 or just col2 is not helped by the index.