This is completely dependent on other factors. Weight loss is dependent on a hell of a lot of things in tandem. It is mathematically strictly better to say, drink only black coffee or sugar-free energy drinks in the morning. Caffeine is an appetite suppressant. The difference is that most people don't have the personal willpower to stick to the fasting without a few calories to help them along.
Weight loss in general is precisely mathematical in pacing and it will absolutely make you lose weight faster if you consume less calories.
You are ignoring an important factor: If you don't have enough calories the body will switch to starvation mode, you'll have low energy, be cold, etc.
The net result is that, unless you are actually starving (which is bad for you), you'll actually lose less weight that way!
This is a myth which has been proved to be absolutely false. You are simply wrong if you are basing any conclusions at all of of this.
I'm perfectly willing to be shown that I'm wrong, but I'll need more than just a sentence from you saying so.
You said "proved" - can you show me the proof?
Probably the best and most researched article on this.
Bullet point 4 here: http://www.leangains.com/2010/10/top-ten-fasting-myths-debun... has an article reference.
Basically, you can certainly go into starvation mode, but not in any reasonable amount of time. Skipping meals or even fasting for a whole day will not cause "starvation mode". Prolonged calorie restriction can affect hormones like Leptin and affect energy expenditure.
https://authoritynutrition.com/3-reasons-why-bulletproof-cof...
I guess it might seem specifically OK for people on Keto diets only, but still, it seems worrisome longterm.
The reason you put butter in your coffee instead of a spoonful of say, sugar, is that fat takes a long time to digest and even a fairly small amount of it (a table spoon of butter in your coffee) takes a while to digest. During that time period your hunger sensation is suppressed.
Coffee also promotes weight loss because it speeds up your metabolism. The idea behind buttered coffee is fairly simple. Put enough fat in your coffee so that you don't get hungry and eat a meal and you help extend your natural faste - and your morning coffee helps speed up your metabolism.
http://paleonick.com/articles/Bullet-Proof-Coffee
There are good reasons for drinking buttered coffee even if you're deliberately trying to avoid weight loss i.e you are following a net neutral or net positive caloric intake diet. When you're fasting your body isn't spending energy digesting food which means less of your body's immediately available resources aren't spent digesting food.
Even if you're getting calories from your coffee it doesn't impact your weight loss because you're still eating less calories than your body needs to maintain the weight.