The notions of weak equivalence are not 'the same' in your examples, rather they coincide on a certain class of objects, namely those that are 'cofibrant-fibrant' as mentioned in Arun's comment. Perhaps there is a more general question hidden in yours, however, as in both examples you have one notion of homotopy equivalence based on a cylinder or cocylinder functor and therefore feeling somewhat geometric, and one notion of 'quasi-isomorphism' relative to some functor, that I will write as $F$, to another context / category, $f:A\to B$ being a 'quasi-isomorphism' if $F(f)$ is an isomorphism. This is slightly more general than the context of model categories and is explored in Baues' work on algebraic homotopy.

The notions of weak equivalence are not 'the same' in your examples, rather they coincide on a certain class of objects, namely those that are 'cofibrant-fibrant' as mentioned in Arun's comment. Perhaps there is a more general question hidden in yours, however, as in both examples you have one notion of homotopy equivalence based on a cylinder or cocylinder functor and therefore feeling somewhat geometric, and one notion of 'quasi-isomorphism' relative to some functor, that I will write as $F$, to another context / category, $f:A\to B$ being a 'quasi-isomorphism' if $F(f)$ is an isomorphism. This is then a slightly more general question than yours in the context of model categories and is explored in Baues' work on algebraic homotopy.