This simply can't be answered with a yes or no. Think of it this way...
Why are you fasting?
If your answer to this is to get the benefits of kicking yourself into Ketosis and lose body fat, the fast itself is not important. What's important is that you get into fat-burning mode and are able to tap into your stored body fat. This can be accomplished mainly with the elimination of carbohydrate and even protein for a prolonged period of time, but isn't inhibited with fat. Therefore, a "fast" with some fat calories isn't going to be detrimental. As long as there is a deficit during the fast, you will see benefits.
If your answer to this is to detox or at least get that feeling of what I like to call "rebooting", the important thing isn't necessarily the Ketosis, but the clearing out of the bowels, improving digestion, cell regeneration and overall just giving the body a break. For me, I'm extremely active, I eat a good amount of calories from different sources and I'd say my digestive function is very good, but also constantly going. When I factfast, I do it primarily to give this part of my body a break.
To me, these are two completely different reasons to fast. If your goal is specifically one or the other, that is going to alter the definition of what a "fast" is to you. Most people seem to want the benefits of both, and if that's the case, your best bet is to completely eliminate caloric intake entirely, and even go as far as not putting ANYTHING into your body that will need digesting or even cause the secretion of digestive juices or enzymes.
