It's fairly common knowledge that there are four Greek words for love, but it's not clear from the Wikipedia article whether certain distinctions persist from Ancient Greek to Koine, namely the distinction between ἀγάπη (agape) and φιλία (philia).
A survey of the uses of these words in Johannine literature will be conducted.
Here is a Bible word study showing all of the words in the Greek that were translated into English as "love" in the NASB translation6 of Johannine literature and their frequency of occurrence:
The verb φιλοπρωτεύω appears in the above diagram, and it means "to have a special interest in being in the leading position, wish to be first, like to be leader" (cf. 3 John 9).7 It does not inherently mean "to love," but is often translated as "loves to be first," hence its appearance in this chart (which begins with the English word "love" and works backwards to the Greek). Another common word for love, ἔρως (eros, erotic love), is not used in Johannine literature. The word στοργή (storge, familial love) means "affection" in modern Greek and is included in the link you referenced, but it does not appear in Johannine literature. Its only appearance in the New Testament is as a compound word with φιλέω: φιλόστοργος (philostorgos, loving dearly / mutual love) in Romans 12:10.
A textual example where both ἀγαπάω and φιλέω occur in the same context is in John 21:15-17, which has been covered in another question which you referenced. Addressing the use of both verbs in this context as well as in several others, user swasheck writes:
There appears to be little difference in meaning between the two words.
1 William Arndt, Frederick W. Danker and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (Chicago: University of Chicago Press, 2000), 6.
2 I obtained these results (and all other search results in this response) by conducting a morphology search in Logos Bible software (v5) for the lemma in Johannine literature within the 27th edition of the Nestle-Aland Greek New Testament, utilizing Logos Greek Morphology.
3 Arndt, Danker, and Bauer, 7.
4 I compared the NET, NKJV, and ESV translations of these verses (here and for all other English translation comparisons in this response).
5 Arndt, Danker, and Bauer, 1056.
6 I also checked the ESV and NKJV and had similar results.
7 Arndt, Danker, and Bauer, 1058.
Within Johannine literature, there seems to be quite a bit of overlap in the uses of the agapao and phileo word groups. Here are a number of pairings of verses across the body of work wherein there seems to be no distinguishable difference of usage. Each pairing below begins with a verse using the agapao word group followed by a similar one using the phileo word group.
Jesus' love for the beloved disciple:
The Father's love for the Son:
The love of sinful things:
The Father's love of Jesus' disciples:
Given these pairings, there is not warrant for a clear distinction between the two terms or word groups within John's writings. Of course, there may still be distinctions in some cases, but these distinctions are better teased out in context. It's not enough to know simply which word was used. For this reason, I leave aside the uses of both word groups in Peter's reinstatement; though personally I think the uses there also point in the same direction as the other pairings here.
All quotations from ESV