Dairy products, especially milk, whey, and yogurt, are insulin secretagogues: they stimulate pancreatic release of insulin. The effect is likely due to amino acids and/or polypeptides in dairy products.  Butter's insulin-triggering effect, doubling or tripling insulin responses (postprandial area-under-the-curve). If insulin is triggered, fat gain follows. Butyric acid causes an insulin response, as does casein. I believe insulin drives the rate of ageing, and this would certainly make me scale back my butter consumption.

I don't see where a moderate amount of butter, when there is a limited or low availability of glucose sources will make you fat, expecially if you are VLC or on a ketogenic diet. I can see where it could be a problem if you were eating 150 or more grams of carbs per day and have metabolic syndrome. Then high levels of insulin would spike even higher prompting the liver to pump out more glucose, and the fat just has no where to go except into storage.

Take your vitamin D3 to get your adiponectin levels up. Take fish oil while limiting Omega-6's to normalize inflammatory responses. Limit carbs, and get the glucose-FA cycle working for you

Protein also stimulates insulin, so protein makes you fat.

Carbs stimulate insulin, so carbs make you fat.

Conclusion: Food makes you fat.

Looking at insulin alone can be very misleading. If glucagon is released along with the insulin ,as happens when eating protein, then the fat storing effects of insulin are negated. 

For low carbers, the study is meaningless as it was done with lots of carbs. I will continue to eat butter and cheese until I see a meaningful study that shows that they cause a real problem.

Fatty acids by themselves don't prompt insulin release. Protein does (but the insulin is balanced out by glucagon), and of course carbs do, but there's nothing inherently dangerous in a fatty acid that would prompt the body to store it away immediately.

Butter delivers higher serum FFAs, indicating that fat might stimulate lipolysis, even though it appears to induce a short term insulin spike.
Carbohydrate might promote a lower insulin response, but it blocks lipolysis.