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.

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EDIT 2023: The above work is published as:

* Michael Barr, John F. Kennison, Robert M. Raphael, _Contractible simplicial objects_, Commentationes Mathematicae Universitatis Carolinae **60** Issue 4 (2019) page 473–495, doi:[10.14712/1213-7243.2019.023](https://doi.org/10.14712/1213-7243.2019.023), [EuDML](https://eudml.org/doc/295068).

with abstract:

> We raise the question of when a simplicial object in a catetgory is deemed contractible. The literature offers three definitions. One is the existence of an “extra degeneracy”, indexed by -1, which does not quite live up to the name. This can be strengthened to a “strong extra degeneracy". Another possibility is that it be homotopic to a constant simplicial object. Despite claims in the literature to the contrary, we show that all three are distinct concepts with strong extra degeneracy implies extra degeneracy implies homotopic to a constant and give explicit examples to show the converses fail.