The confusion, I suspect, is due to using the term fasting. A distinction may be that some people consider you to be in a fast if you skip even one meal and you are certainly fasting if you don't eat for one or more days. Some people can eat little or no carbs (not technically fasting) and be in ketosis.
Ketosis simply means (to me) that I am burning ingested/stored fat (ketones) rather than ingested carbs (glucose.) So, as Jim says, I may or may not be fasting--if I were to eat zero- to very low-carbs (and I can't so I stick to moderate intake) I could eat and still be in ketosis.
If it's noon and I haven't eaten since noon yesterday I am either in ketosis (feeling great) or suffering from inability to release/burn stored fat (feeling weak/tired/hungry) unless I somehow ate 2 days worth of carbs yesterday. I threw in the "mass quantities" part because when I was a binge eater of junk carbs I'm sure I didn't go into ketosis overnight but nowadays I do. I normally eat early enough in the day and few enough carbs that I can't avoid ketosis overnight. I like that because there's no need to eat until it's convenient since I'm unlikely to run out of stored fat to burn in the foreseeable future. In fact, now that I'm acclimated to fat-burning I have to logically decide to stop fasting since I don't get hunger signals. My simple rule is that I don't go longer than 48 hours (4 pm Mon, for example, to 4 pm Wed) without eating so I can be fully active and nourished without risk. Clarification: While I finished eating at 4 Mon, the actual period of ketosis would be more like 24-40 hours since I would have digested and used nutrients from the Mon meal.
The state of "ketosis" is when your body released ketone bodies into your blood and your body begins to make use of stored body fat for energy when there is no glycogen left to produce energy.
As for a "ketosis fast" Ive never actually heard of that...you achieve ketosis by eating and maintaining a VLC diet (most say under 50 g. of carbs a day MAXIMUM, but I say aim lower if ketosis is your goal). This means LOTS of good quality proteins and fats and some leafy greens. You should eventually start heading into ketosis after about 2 days (working out helps, especially if fasted) however you will not be fully keto-adapted for about 2 weeks. When you start to enter into ketosis many have reported an increase occurrence of urination (and a slightly metallic scent), metallic taste in mouth, and increased energy levels and mental clarity.
There is no "ketosis fast" unless you plan on starving yourself and allowing your body to eat away at everything stored...but thats a little counter productive. Try upping your protein and fat and cut almost all carbs, continue to workout if you do and you should notice some of the characteristics of ketosis listed above within a few days.
I'm not really clear on the terminology you're using. You can fast to get into Ketosis and you can fast while in Ketosis. Fasting generally means you don't eat (so it's not a matter of what you can and can't eat). What exactly are you looking to do?
On a fast you typically eat nothing. If your primary goal is to get into ketosis, an 18ish hour fast after a week of low carb eating is a great way to get there. Some choose to extend the ketosis element by eating a few tablespoons of coconut oil to break the fast, providing calories that are easily turned into ketones in the body. This assumes you can handle spoonfuls of coconut oil on an empty stomach; not everyone can.
I think essentially this would be zero-carb, and arguably simply a fat fast. Ketosis can be thought of as how you use fat for fuel when there's no glucose around. So it will occur if you eat nothing (using your bodyfat) or if you eat fat. If you eat anything else, you'll be making glucose available that will be used for fuel first, so disrupting the ketosis.