Influential point

Influence is defined by a comparison, not by appearance: fit the line with the point, fit it again without, and see how much the answer moves. That makes influence a verdict you reach by refitting, unlike the two labels it gets confused with. An outlier in regression has a large residual, a high-leverage point has an $x$ far from $\bar{x}$ (x-bar), and neither one is automatically influential.

Start with six students who study 1, 2, 3, 4, 5 and 6 hours and score 60, 72, 65, 78, 71 and 86, so $\hat{y} = 58 + 4x$ and $r = 0.8107$. Add a seventh who studied 10 hours and scored 75. That point is 23 below the old line, and refitting drops the slope from 4 to 1.6144 and the correlation to 0.5681. Now instead add a seventh who studied 3 hours and scored 100, a residual of 30, larger than the first. Refitting moves the slope only to 3.2742. The bigger miss shifted the line about a third as far, because it sat next to $\bar{x}$ where the line can pivot around it.

So the sentence to drop is "that point has a huge residual, so it is influential." Residual size is not the measure. Distance from the mean of $x$ is what converts a miss into leverage over the whole fit.

Leverage on its own is not the verdict either. Add a seventh student at 10 hours who scored 98, which sits exactly on the original line: that point has more leverage than any other in the set, and the fit stays at $\hat{y} = 58 + 4x$, unchanged to the last digit. It does lift $r$ from 0.8107 to 0.9245, so it is influential for the correlation while having no influence at all on the line. Always say what it is influential for.

Finding an influential point is not permission to delete it. Investigate it, and report the fit both ways if you cannot resolve it.