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Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes.

The standard ("Quillen") projective model structure on $\Ch_{\ge 0}(R)$ has quasi-isomorphisms for weak equivalence, monomorphisms with projective cokernel as cofibrations, and epimorphisms in positive degrees as fibrations. There is also an injective model structure with weak equivalence, monomorphisms in positive degrees, and epimorphisms with injective kernel.

On $\Ch(R)$, there are similar injective/projective model structures, but one drops the "positive degrees" where relevant. There is also a different, "Strøm" model structure on $\Ch(R)$, which is worked out in Chapter 18 of More Concise Algebraic Topology (May–Ponto), see e.g. this MO question. It has homotopy equivalences as weak equivalences, split monomorphisms as cofibrations, and split epimorphisms as fibrations.

Is there an analogue of this "Strøm" model structure for nonnegatively-graded chain complexes? [Wild guess] Perhaps two of them, one with "split epis in positive degrees & split monos" and one with "split monos in positive degrees and split epis"?

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Yes. This is contained in Section 6 of this paper by Christensen and Hovey. In the bounded case, any projective class gives rise to a model structure. In Section 1.4 they introduce a projective class whose weak equivalences are the chain homotopy equivalences. They don't discuss having two such model structures, but using this you could check your "wild guess" by hand.

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