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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.