It appears there is a database on LDS.org that propagates member data in LDS Tools. It appears this same database is used for printing a hard copy member directory in LDS.org and the downloadable database. The database can be downloaded (but not edited) from the Directory section in LDS.org under My Ward, Directory, Export Households.
If member 1 was placed in member 2's household with different name (member 2), then later member 1 either changed into their own household or the household member 2 moved out, member 1 will still have member 2's household name. This is listed in column one (Family Name, aka Household Name) of the exported household database mentioned above.
For some reason, when a person is moved out of a household, into their own household, this database does not update the Family Name column to match the new household name.
In all three; LDS tools, the exported database, and the printed paper member directory, the wrong household will be listed. However, when looking up the member everywhere else it will show the correct household name.