The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition:
Model categories are now required to be complete and cocomplete, whereas Quillen only asked for finite limits and finite colimits.
The two factorisation systems are now required to be functorial.
These changes do make the two definitions genuinely different, since there are non-trivial small model categories in the old sense but not in the new sense.
Question. Are there any textbook results about model categories in the modern sense that are invalid for model categories in the old sense?
I ask because the majority of the textbooks I have looked at so far (e.g. [Hovey, 1999], [Hirschhorn, 2003], [Dwyer, Hirschhorn, Kan, and Smith, 2004]) use the stronger definition. While propositions like "There exists a functorial choice of fibrant/cofibrant replacement" obviously depend on functorial factorisation, it is not so easy to decide whether propositions like "A left adjoint is a left Quillen functor if and only if it preserves cofibrations between cofibrant objects and all trivial cofibrations" [Hirschhorn, 2003, Prop. 8.5.4] needs the stronger definition.
Bonus question. When did people start requiring functorial factorisations, and who if anyone started the trend?
I notice that [Dwyer and Kan, 1980, Function complexes in homotopical algebra] mention functorial factorisation as a nice optional extra, so the idea goes back at least that far.