What is the exact definition used by ArcGIS? Does the entire polygon need to be within X to the target point? Or only at least point within the polygon?
The Spatial Join documentation provides three options for the optional
WITHIN_A_DISTANCE —The features in the join features will be matched if they are within a specified distance of a target feature. Specify a distance in the search_radius parameter. WITHIN_A_DISTANCE_GEODESIC —Same as WITHIN_A_DISTANCE except that geodesic distance is used rather than planar distance. Choose this if your data covers a large geographic extent or the coordinate system of the inputs is unsuitable for distance calculations. WITHIN_A_DISTANCE_3D —The features in the join features will be matched if they are within a specified distance of a target feature in three-dimensional space. Specify a distance in the search_radius parameter.
Neither of your suggested definitions matches the "within a distance of" measurement criteria, since a polygon is a "closed shape defined by a connected sequence of x,y coordinate pairs." Clementini, Felice, and van Oosterom, defined a polygon as an entity composed of the boundary and the interior of a figure. By these definitions, if any part of the figure is within the specified distance, the entire figure is within that distance (even if the figure is multi-part). Your second offering is contrary to the Esri meaning because a point could be exactly equal to the distance, which would match the boundary, but not the interior (not "within the polygon").
Centroids are another matter entirely, since not all definitions need tem to be be within the figure at all.
Generally, if you're not certain of the meaning you can fashion some simple test cases and use them to derive the meaning empirically, since edge cases can be difficult to explain without mathematics.