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](http://www.math.mcgill.ca/rags/seminar/barr021018.html), [slides](http://www.math.mcgill.ca/rags/seminar/Barr-October2018-slides-ConSimpObj-rev.pdf).

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.