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Choice A is correct. A system of two linear equations in two variables, $x$ and $y$, has zero points of intersection if the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, $y=mx+b$, are distinct if the y-coordinates of their y-intercepts, $b$, are different and are parallel if their slopes, $m$, are the same. For the two equations in the given system, $y=2x+10$ and $y=2x-1$, the values of $b$ are $10$ and $-1$, respectively, and the values of $m$ are both $2$. Since the values of $b$ are different, the graphs of these lines have different y-coordinates of the y-intercept and are distinct. Since the values of $m$ are the same, the graphs of these lines have the same slope and are parallel. Therefore, the graphs of the given equations are lines that intersect at zero points in the xy-plane.

Choice B is incorrect. The graphs of a system of two linear equations have exactly one point of intersection if the lines represented by the equations have different slopes. Since the given equations represent lines with the same slope, there is not exactly one intersection point.

Choice C is incorrect. The graphs of a system of two linear equations can never have exactly two intersection points.

Choice D is incorrect. The graphs of a system of two linear equations have infinitely many intersection points when the lines represented by the equations have the same slope and the same y-coordinate of the y-intercept. Since the given equations represent lines with different y-coordinates of their y-intercepts, there are not infinitely many intersection points.