I am not sure what you are calling a discrepancy. For a Database to guaranty the assumption in Halpin's definition, a database must guaranty all business rules and relevant facts are known, and followed.
Say we are going to insert a row in a database that has a foreign key, for Halpin's assumption to hold the database must verify that the foreign key is present and valid in the data set. If it doesn't we have violated the assumption. Same for the transaction quote you mention.
It is an assumption you make can make because it is a guaranty your database provides.
The discrepancy is that Halpin's use of the phrase relates to the correspondence between the real world and the database - to the modelling relationship itself, whereas Meijer and Bierman's use solely relates to the valid states of the database - separate from any meaning that might be taken.
You highlighted the element of referential integrity within their description, whereas I highlighted atomicity. But both are just operational characteristics of the database.
Halpin's CWA comes into play even within a database that is not being updated and has only one relation. Eg. Consider a relation P that models the fact type that person with NAME was born on DOB.
P = { {NAME: Bob, DOB: 2/2/22} }
CWA answers the query "Was Mary born on 2/2/22?" with False.
Say we are going to insert a row in a database that has a foreign key, for Halpin's assumption to hold the database must verify that the foreign key is present and valid in the data set. If it doesn't we have violated the assumption. Same for the transaction quote you mention.
It is an assumption you make can make because it is a guaranty your database provides.