I think that for any size k less than the total size of the database, it is not anonymous. In cases like this, an overly strict definition favoring privacy is the only way to protect people. Similar to how we call 17 year olds children and treat them as such under law even though a 17 year old is far closer to an 18 year old than they are to a 5 year old (yes, there are some exceptions, but these are all explicitly called out). Another example of such an extreme is concerning falsifying data or making false statements. Even a single such statement, regardless of the number of true statements, destroys credibility once found when trust is extremely important. This is why even a single such statement can get one found in contempt of court or destroy a scientist's entire career (and even cast doubt on peers who were innocent).
Overall it is quite messy because it is a mix of a technical problem with a people problem.