Hexagonal unit cell vs crystal system
I this, you are totally confuse with meaning hexagonal unit cell and hexagonal crystal system. First of all, I make you to clear about these two terms. 

Hexagonal unit cell is single entity of hexagonal crystal system. It means there are a no. of repetitive hexagonal unit cells inside a hexagonal crystal system. 

Yes, the diagram given in the question is of hcp i.e. hexagonal unit cell. 

But, your next statement : In the unit cell of hcp there are 17 atoms whereas in unit cell of hexagonal crystal system there are 8.Is this true?

is incorrect. One thing I want to clear you that as per saying there are 17 atoms in unit cell of hcp is wrong. We can make the statement after correction is : To make a unit cell of hcp, we need 17 atoms. It means that there are 17 atoms present in unit cell hcp but to make a unit cell of hcp we require 17 atoms. 

Now, we calculate the no. of atoms present inside the unit cell of hcp or we can say this as no. of effective atoms of unit cell of hcp. 

Calculation of no. of effective atoms of unit cell of hcp : 

No. of effective atoms of unit cell of hcp = 3 (there are a total of three atoms present inside the unit cell of hcp) * 1 (as all three atoms are in main unit cell of hcp, no part of atom is outside the unit cell of hcp) + 2 (there are two atoms present a centre of each base of hexagonal unit cell) * 1/2 ( half part of each atom is inside the main hexagonal unit cell) + 12 (there are a total of 12 atoms present at 12 corners of each base of hexagonal unit cell) * 1/6 ( one-sixth part of each atom is inside the main hexagonal unit cell) = 6 (not 8)