The commonly held understanding is partially correct, but overstated. In reality, the semantic range of the words is broader.
It is true that the words ἀγαπάω (agapaó) and ἀγάπη (agapé) came to mean something like "the highest form of love," but this was primarily due later Christian usage of the term. At the time of the New Testament's writing, the word had a broad semantic range, not all that different than the modern English word love. A Greek-English Lexicon of the New Testament (BDAG), lists the following meanings (italics are suggested glosses; definitions [numbers] are direct quotes; usages [letters] are summaries):
to have warm regard for and interest in another, cherish, have affection for, love
a. by humans - toward other humans (in a variety of fashions) or toward God/Jesus as an act of special devotion
b. by transcendent beings - toward ordinary humans or toward other transcendent beings
to have high esteem for or satisfaction with something, take pleasure in
- to practice/express love, prove one's love
the quality of warm regard for and interest in another, esteem, affection, regard, love
a. of human love - without an object (i.e. an abstract trait), toward an impersonal object, toward human beings, toward God/Jesus
b. of God/Jesus - toward humans, referring to the relationship between God and Christ
a common meal eat by early Christians in connection with their worship, for the purpose of fostering and expressing mutual affection and concern, fellowship meal, a love-feast
From these definitions, it should be clear that it is not a technical term reserved exclusively for "Christian love." This can also be seen plainly by its use in Luke 6:32:
If you love (ἀγαπᾶτε) those who love (ἀγαπῶντας) you, what benefit is that to you? For even sinners love (ἀγαπῶσιν) those who love (ἀγαπῶντας) them. (ESV)
If "sinners" are capable of this love, it cannot be something reserved to true believers.
Additionally, BDAG lists various non-Biblical usages of the verb to describe, for instance, love for fellow humans. The noun form is rare outside of the Bible, although sufficient examples exist to show it did not generally have a special religious significance. The reason it is rare is that it had a "colloquial flavor," which written works generally seek to avoid.
The words φιλέω (phileó) and φίλος (philos) have a similar semantic range. Indeed, φιλέω was the normal verb for many kinds of love in ancient times, but gradually lost ground to ἀγαπάω in most contexts. It does not, however, normally refer to romantic love. BDAG lists the following meanings:
to have a special interest in someone or something, frequently with focus on close association, have affection for, like, consider someone a friend
a. in regards to people - usually literal relatives extra-Biblically; to describe the relationship between believers in the Bible; occasionally, God's relationship to believers
b. in regards to things
c. informally to mean "like/love to do something"
to kiss as a special indication of affection, kiss
pertaining to having a special interest in someone, beloved, dear, and actively loving, kindly disposed, devoted
substantively, one who is on intimate terms or in close association with another,
a. friend - literally or "in a special sense" (for example, fulfilling a specific role - see John 3:29)
b. woman friend (similar to the way a contemporary woman might use "girlfriends")
Eros and Storge
Ancient Greek has two other word sets sometimes associated the word English word love: ἐράω (eraō) / ἔρως (érōs) and στέργω (stérgō) / στοργή (storgḗ). Neither occurs in the New Testament, so the BDAG entries are short. (The words do appear in the Septuagint, however.) The entries in A Greek-English Lexicon (LSJ) list the following meanings:
love, property of the sexual passion; to be in love with
without sexual reference, love warmly
love or desire passionately
a. love, mostly of the sexual passion; love for one; generally, love of a thing; desires
b. object of love or desire
c. passionate joy
the god of love
at Nicaea, a funeral wreath
name of the κλῆρος Ἀφροδίτης
a. love, feel affection, freq. of the mutual love of parents and children; also used in similar authority-subject relationships such as ruler-people or master-dog
b. less freq. of the love of husband and wife; also other near equal relationships such as siblings or friends
c. seldom of sexual love
generally, to be fond of, show affection for
to be content or satisfied, acquiesce
love, affection, esp. of parents and children
rarely sexual love
As the above entries show, these words too have broad semantic ranges.
Typically the four words are described in Christian popular literature something along the lines of:
- agape = Christian love
- philia = brotherly love or friendship
- eros = romantic or sexual love
- storage = family love
These definitions are not incorrect per se, but greatly oversimplify the semantic range of all four words. Essentially, they equate a word with its most common meaning (in Christian literature), which is not proper - neither English or Greek words can be so narrowly defined. For example, agape does indeed typically refer to Christ-like love in the New Testament. However, it does so not because of the lexical specificity of the word, but rather only because the New Testament authors typically use it that way.
To directly address the heart of the question - no, one cannot draw exegetical conclusions from word choice alone. It provides a clue, of course, but context is far word important in determining what the author meant in a specific Bible verse than word choice. The dictionary definition of use of ἀγαπάω/ἀγάπη and φιλέω/φίλος have considerable overlap.
The Greeks did not have four words to describe four different kinds of love. They had four words for one set of complex human emotion with different connotations, just like English has many (overlapping) words that describe aspects of "love".