If Fred says, "The test scores of the group trained by method A is greater than that by method B at a 98% significance level, and therefore method A should be preferred", and Sheila says, "The Bayes factor between hypothesis M1, which assumes that method A and method B produce a similar distribution of test results, and M2, which predicts superior results from A, is 1:38, suggesting that method A is superior" ... they don't actually disagree. Both Fred and Sheila would recommend training by method A.

It's not a traditional dispute about definitions, but (for example) Sheila sniping at Fred for using frequentist methods would be inappropriate. If he genuinely deserves criticism, she will not need to wait long for an occasion where he is wrong.