Imagine any regression line fitted to some data.
Now imagine an extra data point, an outlier some distance away from the main body of the data, but one which lies somewhere along that regression line.
If the regression line were to be refitted, the coefficients would not change. Conversely, deleting the extra outlier would have zero influence on the coefficients.
So, an outlier or leverage point would have zero influence if it were perfectly consistent with the rest of the data and the model that rest implies.
For "line" read "plane" or "hyperplane" if desired, but the simplest example of two variables and a scatter plot is enough here.
However, as you are fond of definitions -- often, it seems, tending to read too much into them -- here is my favourite definition of outliers:
"Outliers are sample values that cause surprise in relation to the majority of the sample" (W.N. Venables and B.D. Ripley. 2002. Modern applied
statistics with S. New York: Springer, p.119).
Crucially, surprise is in the mind of the beholder and is dependent on some tacit or explicit model of the data. There may be another model under which
the outlier is not surprising at all, say if the data really are lognormal or gamma rather than normal.
P.S. I don't think that leverage points necessarily lack neighbouring observations. For example, they may occur in pairs.