Actually, there is a slightly shorter argument: Marc Olschok already proves in his PhD that, in this case, all objects of the left determined one are fibrant. So the left determined one has the same cofibration and the same fibrant objects than Quillen's, so the two model structures are equal.

I mean: every trivial cofibration of the left determined one is a trivial cofibration of Quillen's, so dually, every fibrant of the Quillen's is fibrant in the left determined one, i.e. all spaces are fibrant in the left determined one. BTW, the same argument also proves that with Vopenka's principle, every combinatorial model category such that all objects are fibrant is left determined, fact that I had never noticed before asking this question.

Maybe I should say that I asked this question because I am trying to understand what the homotopy exchange property means for topological spaces. This notion is introduced by Marc Olschok in his PhD to generalize Lafont-Metayer-Worytkiewicz's construction of the folk model structures on globular $\omega$-categories. Marc told me that the homotopy exchange property is satisfied by the cylinder of topological spaces. So unless I am missing something, that implies that the Quillen model structure on delta-generated spaces is left determined, which is very weird.

Yes, the only hypothesis is $a \oplus (-a)$ initial.