"Love" or "Agape love" in John 21:15 et al.?

It is often claimed that there are different kinds of love in Greek and that "agape love" is a special kind of love which refers to the highest and noblest form of love.  People who believe this criticize the KJV and other translations for translating "agape" simply as "love."  D. A. Carson, however, says,

Although it is doubtless true that the entire range of αγαπάω (agapao, to love) and the entire range of φιλέω (phileo, 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 αγαπάω (agapao, to love) and the cognate ἀγάπη (agape, love) can refer to Amnon's incestuous rape of his half sister Tamar (2 Sam. 13:15, LXX). (Exegetical Fallacies, p. 31)

Moreover, backslidden Demas is said to have loved the world with "agape love" in 2 Timotαγαπα) the Son, and hath given all things into his hand," but John 5:20 says, "For the Father loveth (φιλει) the Son, and sheweth him all things that himself doeth."  The Father's love for the Son, which is the greatest love in existence, is described as "agape love" in one instance and "phileo love" in another by the same author John.  There really is no strict rule as to what kind of love each of these Greek words refers to.  Context is key.

Imagine the following dialogue:
  • Betty: "Do you love this actor as much as I do?"
  • Jane: "Of course, I absolutely admire him."
Imagine that a foreign scholar 2000 years from now discovers this dialogue and comes up with the following explanation:
  • In English, "love" referred to the affection found between people who were in close relationships.  "Admire," being the verb form of "admiration," referred to the feeling of esteeming something of great value.  Therefore, Jane thought of the actor as something very valuable, but Betty felt a greater love for him and was perhaps engaged in a relationship with him.
Such an explanation seems absurd, but we make the same exegetical fallacy when we split hairs on the meanings of the words agape and phileo, which were often used interchangeably.