The OP weighed the dice in their own favor by the very phrasing of their question: "What were the writers of the Bible telling Christians when they wrote..."
Slow down a minute. You're referring to two writers, the apostle Paul and "the Elder," that's who claims to be writing 2 John, which apparently means "[John] the Elder," a disciple (not apostle) that the church fathers Papias and Polycarp learned from directly or indirectly. Who are these authors writing to? Paul writes to the Corinthian church, whereas "[John] the Elder" writes to "the lady chosen by God and her children" (NIV).
So the OP is already taking a bit of a leap when they say these writers were writing "to Christians." They're only writing to Christians in general if you interpret the Bible as God-inspired or God-breathed or whatever term you want to use. If you don't, then you have to take the writings at face value, which means Paul wrote to the Corinthian church of the 1st century, and "[John?] the Elder" wrote to whoever "the lady chosen by God and her children" refer to, maybe, and that's a big maybe, the Ephesian church/church in Ephesus (I've read up on this and even in the 2nd century Christians didn't know who the author was or who the author was writing to or about and they've been debating it ever since; heck, the ex-Catholic theologian Alfred Loisy thinks the letter is a forgery and was never actually written to anyone and I'm inclined to believe him).