Sacred ἀγάπη
The idea that the Greek word "agape" (ἀγάπη) means "Godly love" is a very common misconception. Before I elaborate, one of my favorite sayings in hermeneutics is:
Words don't have meaning, people have meaning.
In fact ἀγάπη had a semantic range that included "Godly love," but the semantic range was much broader than this. The actual "meaning" of the word depended on what the author/speaker meant by their use of the word.
Proof:
[Jesus said:] If you love those who love you, what credit is that to you? For even sinners love those who love them. -Luke 6:32
Now look at the verse in Greek:
καὶ εἰ ἀγαπᾶτε τοὺς ἀγαπῶντας ὑμᾶς, ποία ὑμῖν χάρις ἐστίν; καὶ γὰρ οἱ ἁμαρτωλοὶ τοὺς ἀγαπῶντας αὐτοὺς ἀγαπῶσιν. -Westcott / Hort (1881)
Here Jesus uses "agape" four times referring to the kind of love that a sinner has for others. Of course, Jesus is not claiming that a sinner has "perfect, sacrificial, pure, Godly love"... just that they are amiable toward those who are amiable toward them.
English love
The semantic range of the English word "love" also includes Godly love, among other things.
Conclusion
Our use of the English word "love" in 1 Cor. 13 is really no different than Paul's use of the Greek word "agape" in that passage. Both have a semantic range. Both can be used to refer to "Godly love." Both can be used to refer to other things. Regardless of whether you read "agape" or "love" or "perfect, sacrificial, pure, Godly love in the spirit of Christ our Lord" the interpreterreader will need to be careful to remain faithful to authorial intent and understand it in the way it was being used, by the context in which it appears.