What is the difference between monoclinic C and triclinic P lattices?

[Coffee_]

I don't see how for example Monoclinic C and triclinic P are different lattices.

Take a look at monoclinic C :

http://faculty.washington.edu/stenkamp/bstr515/c44.gif

If I'd change one basis vector to go from the origin to the centered point instead I'm actually describing a triclinic P lattice. However they are still given as different Bravais lattices. Where does my mistake/misunderstanding lie here?

 

[read]

In P1:
a≠b≠c, all angles≠π/2, so to two-fold axis

in C2/m
a=b≠c, β≠π/2, γ≠π/2
for primitive basis and you still have 2/m axis going along the bisectrix between a and b