From the first video, it looks like you are doing something close to what I was for the first year and a half I was shooting. You're holding the cue up rather than letting it hang. This causes you to, in an effort to fully follow through, drop the cue as you are following through and thus, the spin on the cue stick. I am by no means an instructor, just looking through these threads for advice for my own weaknesses.
This one can be fixed with the proper drill and a good instructor watching your stroke. A great drill is setting up a straight in shot, marking both the cue ball and the object ball with paper hole reinforcements. Then, behind the cue ball, put 3 more reinforcements, one directly behind and one on both sides of that. Lastly, set up a golf tee on each of the outer paper hole reinforcements. Practice stroking through the middle part of the cue ball (top, center, bottom, doesn't matter, just no side english) while stroking between the tees without hitting them over. Someone posted a link to that drill on youtube but now I can't find it.