I want to understand the definition of a commutative differential graded algebra (CDGA) to be formal.
Actually I encountered two definitions, but I have trouble with both.
A commutative differential graded algebra $A$, again with $A^0 = \mathbb Q$, is called formal if $A$ has a model with vanishing differential. This is equivalent to requiring that the cohomology algebra of $A$ (viewed as a differential algebra with trivial differential) is a model for $A$ (though it does not have to be the minimal model).
From Griffiths, Morgan:
A space is said to be formal if the homotopy type of the DGA of forms on the space is the same as the homotopy type of the cohomology ring of the space. Thus, if $X$ is formal and $\mathcal m_X$ is a minimal model for the forms on $X$, then there is a DGA map $\mathcal m_X\to (H^*(X),d=0))$ which induces the identity on cohomology.
As far as I know, a CDGA is a model for another CDGA, if they are quasi-isomorphic (i.e. isomorphic in cohomology). A CDGA is a minimal model if it is a model and it is minimal (free and decomposable).
So here are my issues with the definitions:
Wikipedia: To me it seems that the complex $(H^*(A),d=0))$ always is a model of $A$.
Griffiths, Morgan: We already have the existence of an isomorphism $\mathcal (H^*(m_X,d_\mathcal m))\to (H^*(X),d=0))$. But this is not strong enough, we need this to be the identity. But in order to speak about the identity, the spaces must be same, which they are only up to isomorphy.
It seems like I missed a crucial point of the definitions. But what is it?