Yes. This is contained in Section 6 of [this][1] 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.


  [1]: https://arxiv.org/abs/math/0011216