The effort of distinguishing between two camps in a "reasonable" way is inspired by my claim that thinking about one camp as pro-free-market and one camp as not is not a reasonable way to distinguish Keynesians and any of a variety of austerians out there.
I think this is perhaps one reasonable way (although of course more mainstream economists than just Keynesians talk about market failures, and you'll have some acknowledgement of them by some austerians). This terminology is misleading too, though, and also biases people into thinking that those of us who talk about "market failures" are not pro-market.
As I've said many times before, I don't like the phrase "market failure" and I rarely use it myself (do a word search on the blog - most times I use it will probably be in reference to others who have already used it). Market failures are really just descriptions of situations where institutional preconditions for the nice, neat functioning of markets don't exist. It seems deeply unfair to me to call that a "market failure", because the market hasn't "failed" at all - it hasn't been given the chance to operate, so why blame it for "failure"?
For similar reasons I hate the terms "public good" and "private good". Goods are goods and can be provided by any number of institutions (see Coase on lighthouses). The salient point is not calling something a "public good". A pure public good is an illusion (just as is, probably, a pure private good - although such a thing is far more conceivable). What matters is that all goods have a certain degree of external costs and benefits, all goods have a certain extent to which they are excludable and non-excludable and rivalrous or non-rivalrous. Talking about this as a dichotomy makes it very easy for people who just plain don't like government to forestall any intelligent discussion of public policy.
Toll roads can be provided privately - case closed! There's no case for government roads because obviously it's not a public good if people can be excluded from it!
No, not exactly - not if you dispense with the public/private dichotomy nonsense and just say "the market can clearly provide lots of this stuff, and it should provide it and we should celebrate that, but the network externalities, the major (external) benefits to future generations, the non-rivalrous nature of most roads, and the cost of complete exclusion suggest that the government and the market can be partners in road construction and there's no point in pretending we have to choose".
So this "market failure" material is definitely a line of contention for a lot of people, but I do think the phrase "market failure" is (1.) simply misleading, and (2.) obscures the fact that there's nothing anti-market about this approach.
The real difference between these groups isn't really how they view markets. Both sides are pro-market. Both sides see an important role for markets in human society. When we do differ, it's unclear why austerians' "when we're holding a hammer everything looks like a nail" mentality makes them more pro-market (explain to me why that doesn't make them less pro-market?). So the real difference isn't over markets. The real difference is over government, and that difference is stark. I think that liberal government is an institution that humans have evolved that solves a lot of their problems - it can be run well or poorly, but it's a good evolved social tool with lots of good uses (just like the market). Contrast that with people who think government is inherently immoral - or (just as one example) with Bob Murphy who won't even endorse raising a tax to prevent the imminent destruction of the human race.
That's where the real difference of opinion is - over government, not markets.