This happened with model categories, where Quillen's original definition only required the existence of finite limits and colimits as well as a factorization of all maps that wasn't necessarily functorial. Almost all books today take model category to mean what Quillen called a closed model category (and in fact, they go one step further. They add functorial factorization as well), since it makes the proofs easier and the conclusions much more far-reaching.
This is at the cost of losing some categories of some kinds of finite chain complexes (I have never run into one of these in practice, but I suppose that some people do) as model categories, but this stronger definition includes almost every usage of model-category theory in homotopy theory.
I will note, however, that while model categories are almost always taken to be closed (outside of Quillen's original paper), functorial factorization is not nearly as standard (if I remember correctly, Jacob Lurie doesn't require functorial factorization in his definition in Higher Topos Theory).
For more, you can read the introductions to:
Model Categories by Mark Hovey
Model Categories and their Localizations by Phil Hirschhorn
Homotopy Limit Functors on Model Categories and Homotopical Categories by Dwyer, Hirschhorn, Kan, and Smith
