The Greek word ὁ is much more than what we call the “definite article.” Depending on context, it can function as a demonstrative or a possessive pronoun. In this verse, it is called an “anaphoric” article.
As you get into Greek, you will find that the grammar is not always correct. Sometimes, like in Revelation, the grammar can be explained by the fact that John is in an ecstatic state and the grammar irregularities enhance the message. But even in non-apocalyptic literature, you will find that the grammar rules are not always followed. This should not affect your view of inspiration. All good writers violate grammar, periodically, for effect, or for other reasons.
Since a word in one language does not line up exactly with a word in another language, translation is often a matter of choosing one meaning or the other, which generally means some information is lost. It also means all translations are interpretive. Words have what is called a "semantic range," a breadth of meaning. They don't have one "literal" meaning. In this passage, is Paul saying Timothy is his legitimate child or genuine child? Is Paul talking about Timothy's legal status or character?