Can you give the grammatical rule that distinguishes between these two cases?

The problem with the word “that” is that it has many different uses. It can be a conjunction, a determiner, a relative pronoun, a demonstrative pronoun, and even an adverb.

In these examples you’ve provided, “that” is being used as a relative pronoun. Many grammarians consider all “that” deletion to be “informal”, but there are still rules. In this case, the rule is that it can be deleted when it is the object in its relative clause, but not when it is the subject.

In the first sentence, the relative clause is “that Mary saw” and it is the object (Mary is the subject; English inverts word order when the relative pronoun is an object). In the second sentence, “that had a white collar” is the relative clause and it is the subject (“collar” is the object of “had” here). We can delete it when it is the object, but not the subject.

Well done! Is this something you were formally taught? Even when I saw the explanation, it took me a while to piece it together. Subjects and objects of relative clauses was something I had not previously given any thought to.

@Demosthenes I don’t remember being formally taught it until college when I took a course on English Grammar, but I started learning most of these rules when I took Latin in high school (which helped inform my understanding of English grammar, despite the significant differences between the two languages).