What do we mean by contractible for simplicial objects in a category?

I don't have time to wade through terminology, but this seems to be an exercise in avoiding the use of simplicial homotopies, as defined for example in Definition 5.1 on page 12 of Simplicial Objects in Algebraic Topology. These homotopies make sense for simplicial objects in any ambient category $\mathcal{C}$ whatsoever, so that with this notion there is no such thing as a $\mathcal{C}$ "with insufficient structure to support homotopies''. Extra degeneracies give a particularly convenient way to construct just such a homotopy. If you have a terminal object, then you have a trivial map $\epsilon$ from any simplicial object $X$ to the constant simplicial object $\ast$ at the terminal object. A "base point'' is a map $\eta$ from $\ast$ to $X$ and is determined by a map in $\mathcal{C}$ from the terminal object to $X_0$. Then $\epsilon \circ \eta$ is the identity on $\ast$ and we say that $X$ is contractible if $\eta\circ \epsilon$ is homotopic to the identity. That makes sense in any $\mathcal{C}$, and it agrees with the usual notion in the usual examples.

Actually the issue is not that simple. Here is a talk by Mike Barr on showing there are three different notions of 'contractible' for simplicial objects in a category that do not coincide even for simplicial sets. They even give explicit examples.

Michael Barr (joint with John F. Kennison, R. Raphael), Contractible simplicial objects, talk 9 October 2018 in the Logic and Categories seminar, McGill University, abstract, slides.

They find "Strong extra degeneracies" implies "Extra degeneracies" implies "Homotopic to a constant simplicial object", and these are strict implications. See the slides for precise definitions. Beware that the 'extra degeneracies' given in my question are not necessarily the same as Barr et al's definition.