Some things preach well, and then some things preach very well! The idea that there are four unique and distinct words for “love” in Koine Greek is very popular among evangelical pastors. I surmise one reason it is so widely accepted as true, in light of the textual issues, is that it preaches well.
No one has had more influence in the mass proliferation of the four loves than C.S. Lewis. Lewis lays out this four-fold study of the Greek words for love in his book The Four Loves. Lewis argues that the four Greek words for love (agape, phileo, eros, storge) all communicate different types of love (unconditional, friendship, romantic, affectionate). But is this true? D.A. Carson thinks not and addresses this issue in his book Exegetical Fallacies:
In a similar vein, although it is doubtless true that the entire range of ἀγαπάω (agapaō, to love) and the entire range of φιλέω (phileō, to love) are not exactly the same, nevertheless they enjoy substantial overlap; and where they overlap, appeal to a “root meaning” in order to discern a difference is fallacious. In 2 Samuel 13 (LXX), both ἀγαπάω (agapaō, to love) and the cognate ἀγάπη (agapē, love) can refer to Amnon’s incestuous rape of his half sister Tamar (2 Sam. 13:15, LXX). When we read that Demas forsook Paul because he loved this present, evil world, there is no linguistic reason to be surprised that the verb is ἀγαπάω (agapaō, 2 Tim. 4:10). John 3:35 records that the Father loves the Son and uses the verb ἀγαπάω (agapaō); John 5:20 repeats the thought, but uses φιλέω (phileō)—without any discernible shift in meaning. The false assumptions surrounding this pair of words are ubiquitous; and so I shall return to them again. My only point here is that there is nothing intrinsic to the verb ἀγαπάω (agapaō) or the noun ἀγάπη (agapē) to prove its real meaning or hidden meaning refers to some special kind of love.
D.A. Carson. Exegetical Fallacies, 32-33.