Feature To Point tool is the right one to use. By design, If the Inside option on the dialog box is unchecked, for an input polygon feature the output point will be located at the center of gravity (centroid) of the polygon. Depending on the shape of the polygon, this centroid may or may not be inside the polygon. If you want to make sure to get a point inside the polygon, you should check the Indise box, then the point will be inside, but not necessarily at the "center".
Again, depending on polygon shape, for the Inside option, it may not be always true that the point is both inside and at the center.
We made some improvement on the location of the point for the Inside option in ArcGIS 10. You may see the differences between 9.3.1 result and 10 result; hopefully the 10 result is better overall. So for the comparison, you should use the same software version to process original polygons and the moved/changed polygons using the same Inside or Centroid option. If no shape change happens, the resulting points should be at the same inside or centroid locations relative to the polygons. If the polygon shape changed, then the point locations will change; it may not be a reliable comparison.
Help this helps you better understand the tool.