Witt vector question

Question

I've started reading various papers and notes on Schemes over the Witt Vectors. In example 8.8 of these: https://www.uni-due.de/~mat903/books/esvibuch.pdf W2 has addition defined as $k \oplus k\cdot p$ for an addition group. In definition 4.1 of this http://arxiv.org/pdf/1301.0857.pdf, and in some others, addition is defined for $W_2(k)$ by $$S_0 = a_0 + b_0,\quad S_1 = a_1 + b_1 - \sum_{0<i<p}\frac{1}{p}\binom{p}{i}a_0^i b_0^{p-i}.$$ I've tried writing out some simple examples, but am still confused. How does these two definitions compare?