The notational part of this debate is definitely not agreed upon. Andrew Gelman and Christian Robert both argue for p(data|H_0) in comments on those blog posts.

I think this comment on that blog says it all:

>I fully agree with Andrew that, from an objective perspective, there is no misunderstanding whatsoever. Unless you get into arcane measure theory details, conditioning upon a value and taking this value as fixed amount to the same thing. Discussing about notations does not seem like an optimal use of our time (times?)…

There simply is no difference between conditioning on a value chosen before hand vs treating it as fixed. If you imagine a Venn diagram of the two scenarios, the event space under question is exactly the same. Whether your conditioning value can actually change (a rv) or whether it is universally constant, you are still only considering the space where your conditioned value is fixed.