How to specify spatial join relationship "within or partially within"

Question

Using ArcGIS Pro 3.4, I'm trying to use Spatial Join to identify which counties are "within or partially within" a congressional district, but exclude those that merely touch a boundary. For example, Chaves county (yellow polygon) should be in multiple districts (black boundary lines), while Deaf Smith should be in only one district.

Using "largest overlap" doesn't work because it only returns the single largest overlap and "intersect" doesn't work because shared boundaries create many incorrect duplicates.

SHPs available here - https://www.census.gov/geographies/mapping-files/time-series/geo/cartographic-boundary.html

Join Features is supposed to accomplish this using "Overlap" but is still returning polygons with shared boundaries. (The bottom image is directly from this link and shows that shared boundaries should NOT be included.)

This is the result of Join Features using "Overlaps" and one-to-many, four counties returned rather than just the one:

MikeW · Answer

The data is directly from public sources and they should be topologically aligned. Do folks have examples of datasets where the Join Features "Overlaps" relationship works as described?

DavidPike · Answer

I think you've got some slithers of overlaps from the datasets.  Certainly recommend Jack's approach, and possibly Tabulate Intersection (Analysis)—ArcGIS Pro | Documentation might help you to figure out what's going on with the potential overlaps - or may be used to achieve the join instead (remove <1% overlap etc).

danretired · Answer

perhaps have a look at the Select by location options, there are some cases not covered by spatial join.

Select By Location graphic examples—ArcGIS Pro | Documentation