definition of scoring function Score functions are used in the evaluation of probabilistic forecasts. The question: is the score function positive oriented or negative oriented?
For example, in the paper Strictly Proper Scoring Rules, Prediction, and Estimation it says that $S(Q,Q) \geq S(P,Q)$ for any distributions $P$ and $Q$,
Conversely, in the paper Predictive Model Assessment for Count Data it is the opposite, i.e. $S(Q,Q) \leq S(P,Q)$ for any distributions $P$ and $Q$.
What is the most common way to define a score?

I think it’s useful here to refer back to wikipedia.
The first equation you quote above is for a proper scoring rule. This is one where the highest possible score is reported by using the true probability distribution.
I can’t access the second document referenced in the question so I can only guess that it is a) a typo or b) the score is defined such that negative values are optimal.
