I am not sure that manufacturers all use the term correctly, so that muddies things even more than they're already muddied.
In my own mind (and I could be completely wrong) --
A skinny is a pant (usually a jean) that is very slim all the way down the leg, and may have scrunch (i.e. extra length) on the bottom. Increasingly, these pants are literally skin-tight -- but that is not a necessary part of the definition. The skin tight ones tend to run a little bit shorter, for a look that gives little to no scrunch. Depending on your point of view, your body type, your fashion preferences, and how to intend to wear it, this look is either "sleek" or "calf-sucking."
Cigarette pants emerged much earlier. Earlier versions were all side zip, I believe -- though that is not true any more. In my imagination, a cigarette pant is also very slim but perhaps not skin tight, and it tapers gradually to the bottom so the ankle area is quite slim. Often these are ankle pants.
Slim pants? Who knows? Those could be anything from straight legs to skinnies to cigarette.
