A point has high leverage when its x value is far from the mean of x, regardless of where its y sits. Leverage measures opportunity to influence the fit, not whether the point took the opportunity. In the high leverage preset above, the far right point has leverage near 1, meaning the fitted line is nearly obliged to pass through it.
Cook's distance combines both: how unusual the point is in x, and how badly the model misses it in y. A point can have high leverage and almost no influence if it happens to sit right on the trend. The conventional flag is a Cook's distance above 4/n, which is what the dashed ring marks.
Turn on squared residuals and the name stops being jargon. The fitted line is the one position where the total area of those red squares is as small as it can be. Squaring is why one distant point matters so much more than several nearby ones: doubling a residual quadruples its contribution.
