There are lots of benefits you get from fasting, some of them you get just from protein/carbohydrate restriction, some of them you don't. Whether or not you call it protein/carb restriction "fast" or not, is a purely verbal matter. There are relevant differences and similarities between eating only fat and not eating anything. People have, however, disagreed about which is the real definition here and here.

As to the actual question about autophagy. You'll still get autophagy from protein restriction alone, since the amino acids have to come from somewhere. Carbohydrate would reduce autophagy I strongly suspect, given that it'll spare protein. Eating adequate carbohydrate but restricting protein, you'd be doing something very unlike a fast (though still valuable). Eating plain cream or some such, would plausibly reduce autophagy in the short term, as it does increase insulin which does inhibit autophagy. I would imagine, given that even thinking about food increases insulin, that pure fat would so the same, but whether it does so enough for you to care is up to you. If eating some fat allows you to extend a fast, then I can only assume you're gaining a net benefit.