In the comments on that last post, Marc Benitah asked why I thought this was such an important debate. I had been planning to explain this anyway, so I'm glad he asked.
I think the distinction between "protectionism" and "terms of trade manipulation," as possible objectives of trade rules, is important because of the scope each one sets for the trade regime. With anti-protectionism as its goal, the system can use traditional and familiar provisions such as GATT Article III and TBT Agreement Article 2.1 to identify and prohibit protectionist measures. I see this approach as widely accepted and legitimate, and the core of the trade regime. (Of course, we can argue all day about what exactly Article III means and what protectionism is. Is it about intent or effect? What exactly do we do with likeness? We'll never agree completely, but there are a range of meanings that are easy enough to understand, and they make the system work pretty well, in my view).
By contrast, with terms of trade manipulation, the scope of the system is a lot less clear to me. Keep in mind, the optimal tariff, I'm pretty sure, is not the only measure the proponents of this view have in mind. They also envision that domestic laws and regulations could be used for terms of trade manipulation. So how would the trading system address such measures through specific legal obligations? How would the Seal Products measure, or the Tuna-Dolphin measure, or plain packaging laws be examined if the issue was whether they constitute terms of trade manipulation? I'm not sure, and that's what worries me about this approach. If they want to use the exact same provisions we have now, and interpret them all in the exact same way, then I guess there's no difference. But I'm not completely sure this is the case.
Johnathan Dingel raises (on twitter) another issue: There are actually some additional theories (pp. 319-341) about the purpose of trade agreements. There's the domestic commitment theory (a trade agreement "can serve as a commitment device for a government to close the door to domestic lobbies"), and several new theories: "“firm-delocation” externalities in the presence of free entry, (ii) “profit-shifting” externalities, and (iii) trade-volume externalities when prices are determined by bilateral bargaining.
My own theory, for which I don't have a name yet and which is probably an oversimplification, is that trade agreements are only about economics to a limited extent. If economists were running things, we all would have eliminated protectionist measures long ago. But while economists have a little input on trade (enough to prevent protectionism from spiralling out of control), trade policy is mostly made by voters and the politicians they elect. As a result, the main force in trade policy is not economics but plain old nationalism: "us" versus "them." For example, people support "Buy America" provisions not because they make Americans better off (although a small number of people may think they do), but because they have the word "America" in them. (How can you vote against America?!?!). And they get fired up about currency manipulation by China, not because they are worried about the Chinese government's build-up of foreign currency reserves, but because they hear the word "China." The only way we can keep this nationalism in check and convince anyone ("us" or "them") to support trade agreements is if we all agree to be less protectionist at the same time. Most people believe "they" are cheating and "we" are not, so trade agreements serve to reassure us that "they" have promised not to cheat and therefore we can relax our nationalist impulses a bit. Thus, trade agreements are as much about international relations and psychology/sociology as they are about economics, although hopefully they do incorporate some good economics (not always the case -- I'm looking at you, anti-dumping!).