The word fast has been changed to mean whatever the person defining the fast says it is. IMO, only a water fast is a fast. And IF isn't fasting, it's just waiting a long time between meals. But that's just me.
Juice fasts aren't fasts because even if it's liquified, fruits and veggies are food. Eating food isn't fasting. Drinking just lemon juice with cayenne pepper and honey isn't fasting, because again, they're foods. So, to me, calling a 300-400 calorie coffee in the morning anything besides breakfast is a bit delusional. But no more delusional than any other things called fasting that I don't personally think are fasting.
All of that said, in the current lexicon of health junkies everywhere, I'm not predisposed to "correct" every person who uses the word fast in a way in which I don't agree. And belittling someone neither elevates the belittler nor does it change the status of the "belittlee." If the Bulletproof executive guy calls butter coffee part of a fast, then in his plan, that is how he has defined fast.
"Right is right, even if no one is doing it; wrong is wrong, even if everyone is doing it." - St. Augustine