Rather than stating the general definition again, we can verify this by walking through what actually happens when we construct an influence line diagram (ILD) for a specific response, such as the reaction at support A of a simply supported beam.
To construct this ILD, we place a unit load at a series of different locations along the span, one at a time, and for each position we calculate the resulting reaction at A. The location we are computing the response for (support A) never moves during this process — what changes from one calculation to the next is only where the unit load sits along the span. Plotting the computed reaction value against the load's position produces the influence line.
The worked example confirms that only the load's position is varied while the point of interest for the response stays fixed throughout.
Therefore, the correct answer is points remain fixed, position of load changes.