The level of systemic inflammation doesn't have anything to do with the raw amount of inflammatory precursors available. It's not like linoleic acid just becomes arachidonic acid and it's not like arachidonic acid just becomes pro-inflammatory eicosanoids and it's not like pro-inflammatory eicosanoids just run wild willy nilly. It's all ridiculous. There is a very high quality randomized controlled trials where the participants supplement with high amounts of arachidonic acid and not only don't change their inflammation levels but maintain completely optimal levels of inflammation. //www.ncbi.nlm.nih.gov/pubmed/22188761
It is well known that having enough omega-3 fats not only modulates the conversion of arachidonic acid to inflammatory molecules but turns off the signaling of already-present ones. This is a Japanese study so I assume that the participants were eating fish.
The evidence that dietary arachidonic acid increases the level of systemic inflammation is very shaky and has been refuted recently. It was mostly in vitro stuff, observational, and stuff like a "low arachidonic acid diet" where they make many changes not just arachidonic acid levels. But as these researchers have shown dietary arachidonic acid has no effect on systemic inflammation in healthy subjects.
Linoleic acid may be best limited, though. Because while it doesn't necessarily equal more inflammatory signaling molecules (that tends to be determined by levels of hormones like insulin, glucagon, leptin, and by the actual need for them as dictated by antigens) it could potentially impair the action of anti-inflammatory signaling molecules, letting the inflammatory response spiral out of control. But I don't think that the proper research has been done, because most of it is like 15g vs. 25g when I would want to know what happened at 5g or so. It certainly isn't more omega-6 of any kind = more inflammation. Potentially oxidative stress and potentially some other interference in biological affairs but not inflammation per se. As for omega-6 being anti-inflammatory, if you are deficient in omega-3 fats it might be the only source of anti-inflammatory fatty acid precursors, like GLA. But that's not really a point in its favor.
A lot of this is speculative but well-grounded in basic knowledge, but what isn't speculative is that arachidonic acid doesn't mean more inflammation.
What is very speculative is that the changes observed in the study you referred to had anything to do with animal products! They didn't even do a dietary analysis, they completely fail the transparency test. We don't even really know what they said to the subjects. Could it be that the "vegetarian" group spontaneously increased their intake of magnesium-rich foods and were less tense because of that or vitamin c or something else while the meat and fish groups just went for more Big Macs and Fish Fillets? We don't know. Baaad science.