Formal can mean slightly different things in different contexts.

A commutative differential graded algebra (CDGA) is formal if it is quasi-isomorphic to it's homology. This is stronger than having all the higher Massey products equal to 0 (I think there are such examples in the Halperin-Stasheff paper).

To a space you can associate a CDGA (via Sullivan's $A_{pl}$ functor) which is basically the deRham complex when the space is a manifold. In nice cases this functor induces an equivalence from the rational homotopy category to the homotopy category of CDGA. Quasi-isormorphic CDGA correspond to (rationally) homotopy equivalent spaces. You can also tensor with the reals to get real CDGA.

If A is a CDGA which is quasi-isomorphic to $A_{pl}(X)$ for a space $X$ then A is often called a model of X. A space is formal if $A_{pl}$ of it is formal. So a formal space is modeled by its cohomology. In that sense its rational homotopy type is a formal consequence of its cohomology.

I think you have to be slightly careful with using $C^*$. This functor lands in differential graded algebra which are not commutative, so possibly the notion of formality could be different. In particular if you consider two CDGA there may be more maps strings of quasi-isomorphisms between them as DGAs then as CDGAs(. I don't know of any examples, but it's a point believe it is unknown if two CDGA that are quasi-isomorphic as DGA have to worry about)be quasi-isomorphic as CDGA.

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Formal can mean slightly different things in different contexts.

A commutative differential graded algebra (CDGA) is formal if it is quasi-isomorphic to it's homology. This is stronger than having all the higher Massey products equal to 0 (I think there are such examples in the Halperin-Stasheff paper).

To a space you can associate a CDGA (via Sullivan's $A_{pl}$ functor) which is basically the deRham complex when the space is a manifold. In nice cases this functor induces an equivalence from the rational homotopy category to the homotopy category of CDGA. Quasi-isormorphic CDGA correspond to (rationally) homotopy equivalent spaces. You can also tensor with the reals to get real CDGA.

If A is a CDGA which is quasi-isomorphic to $A_{pl}(X)$ for a space $X$ then A is often called a model of X. A space is formal if $A_{pl}$ of it is formal. So a formal space is modeled by its cohomology. In that sense its rational homotopy type is a formal consequence of its cohomology.

I think you have to be slightly careful with using $C^*$. This functor lands in differential graded algebra which are not commutative, so possibly the notion of formality could be different. In particular if you consider two CDGA there may be more maps between them as DGAs then as CDGAs (I don't know of any examples, but it's a point to worry about).