Hi Paul,
The Bayes factor in the first example is a ratio of marginal likelihoods. It’s easy to prove; just write the posterior odds out as the ratio of two integrals, and you’ll have to correct for the prior odds in the integral over the negative and positive parameter spaces. The ratio of marginal likelihoods pops out as the factor by which the prior odds are multiplied to obtain the posterior odds.

The point is that there is a difference, but the difference is not in the definition of the Bayes factor — that’s the same — but rather in the models being compared (the RSS team used Figure 3B, and my first example uses Figure 3A). So it’s not a Bayes vs frequentist issue as much as a modelling choice — or, if you like, a difference in the questions you choose to ask.