According to Wikipedia, the word was used rarest in classical Greek, but was used in various ways:
(as a verb) -- to greet with affection
to show affection for the dead
love for spouse or family
I'm not sure the writers of the New Testament meant it in any way other than just "love". If I remember correctly, virtually every use of the word "love" in English is translated from "agape". (In TWI some "teacher" would breathlessly reveal that some instance of love was ...the word agape...as if it was some cosmic truth). I believe that it was retroactively assigned the meaning or interpretation of love from or for God.
Biblical writers and theologians needed to present love that proceeded from God, or manifested by Christians, was somehow different than love manifested by disbelievers. I doubt you could subjectively see any difference between Christians and non-Christians in how they love. Of course any attempt to meaningfully define what God's love entails runs into the problem of any possible unloving action by God spurring a redefinition of love that includes that action.