First, I do not want to come off as knowing NT Greek. I do manage to muddle my way to good information via some resources. Hope you find the stuff below useful.
Yes, both these words are forms of agape.
Here's some good stuff from the excellent resource
Vine’s Complete Expository Dictionary of Old and New Testament Words.
BELOVED
A. Adjective.
agapetos (ἀγαπητός, 27), from agapao, “to love,” is used of Christ as loved by God, e.g., Matt. 3:17; of believers (ditto), e.g., Rom. 1:7; of believers, one of another, 1 Cor. 4:14; often, as a form of address, e.g., 1 Cor. 10:14. Whenever the KJV has “dearly beloved,” the RV has “beloved”; so, “well beloved” in 3 John 1; in 1 John 2:7, KJV, “brethren” (adelphos), the RV has “beloved,” according to the mss. which have agapetos. See DEAR.
B. Verb.
agapao (ἀγαπάω, 25), in its perfect participle passive form, is translated “beloved” in Rom. 9:25; Eph. 1:6; Col. 3:12; 1 Thess. 1:4; 2 Thess. 2:13. In Jude 1 the best texts have this verb (RV); the KJV, “sanctified” follows those which have hagiazo. See LOVE.
Note: In Luke 9:35, the RV, translating from the most authentic mss., has “My chosen” (eklego), for KJV, “beloved” (agapetos); so in Philem. 2, “sister” (adelphe).
LOVE (Noun and Verb)
A. Verbs.
1. agapao (ἀγαπάω, 25) and the corresponding noun agape (B, No. 1 below) present “the characteristic word of Christianity, and since the Spirit of revelation has used it to express ideas previously unknown, inquiry into its use, whether in Greek literature or in the Septuagint, throws but little light upon its distinctive meaning in the NT. Cf, however, Lev. 19:18; Deut. 6:5.
“Agapē and agapaō are used in the NT (a) to describe the attitude of God toward His Son, John 17:26; the human race, generally, John 3:16; Rom. 5:8, and to such as believe on the Lord Jesus Christ, particularly, John 14:21; (b) to convey His will to His children concerning their attitude one toward another, John 13:34, and toward all men, 1 Thess. 3:12; 1 Cor. 16:14; 2 Pet. 1:7; (c) to express the essential nature of God, 1 John 4:8.
“Love can be known only from the actions it prompts. God’s love is seen in the gift of His Son, 1 John 4:9, 10. But obviously this is not the love of complacency, or affection, that is, it was not drawn out by any excellency in its objects, Rom. 5:8. It was an exercise of the divine will in deliberate choice, made without assignable cause save that which lies in the nature of God Himself, Cf. Deut. 7:7, 8.
“Love had its perfect expression among men in the Lord Jesus Christ, 2 Cor. 5:14; Eph. 2:4; 3:19; 5:2; Christian love is the fruit of His Spirit in the Christian, Gal. 5:22.
“Christian love has God for its primary object, and expresses itself first of all in implicit obedience to His commandments, John 14:15, 21, 23; 15:10; 1 John 2:5; 5:3; 2 John 6. Self-will, that is, self-pleasing, is the negation of love to God.
“Christian love, whether exercised toward the brethren, or toward men generally, is not an impulse from the feelings, it does not always run with the natural inclinations, nor does it spend itself only upon those for whom some affinity is discovered. Love seeks the welfare of all, Rom. 15:2, and works no ill to any, 13:8–10; love seeks opportunity to do good to ‘all men, and especially toward them that are of the household of the faith,’ Gal. 6:10. See further 1 Cor. 13 and Col. 3:12–14.”*
In respect of agapao as used of God, it expresses the deep and constant “love” and interest of a perfect Being towards entirely unworthy objects, producing and fostering a reverential “love” in them towards the Giver, and a practical “love” towards those who are partakers of the same, and a desire to help others to seek the Giver. See BELOVED.
2. phileo (φιλέω, 5368) is to be distinguished from agapao in this, that phileo more nearly represents “tender affection.” The two words are used for the “love” of the Father for the Son, John 3:35 (No. 1), and 5:20 (No. 2); for the believer, 14:21 (No. 1) and 16:27 (No. 2); both, of Christ’s “love” for a certain disciple, 13:23 (No. 1), and 20:2 (No. 2). Yet the distinction between the two verbs remains, and they are never used indiscriminately in the same passage; if each is used with reference to the same objects, as just mentioned, each word retains its distinctive and essential character.
Phileo is never used in a command to men to “love” God; it is, however, used as a warning in 1 Cor. 16:22; agapao is used instead, e.g., Matt. 22:37; Luke 10:27; Rom. 8:28; 1 Cor. 8:3; 1 Pet. 1:8; 1 John 4:21. The distinction between the two verbs finds a conspicuous instance in the narrative of John 21:15–17. The context itself indicates that agapao in the first two questions suggests the “love” that values and esteems (cf. Rev. 12:11). It is an unselfish “love,” ready to serve. The use of phileo in Peter’s answers and the Lord’s third question, conveys the thought of cherishing the Object above all else, of manifesting an affection characterized by constancy, from the motive of the highest veneration. See also Trench, Syn., Sec.xii.
Again, to “love” (phileo) life, from an undue desire to preserve it, forgetful of the real object of living, meets with the Lord’s reproof, John 12:25. On the contrary, to “love” life (agapao) as used in 1 Pet. 3:10, is to consult the true interests of living. Here the word phileo would be quite inappropriate.
Note: In Mark 12:38, KJV, thelo, “to wish,” is translated “love” (RV, “desire”).
B. Nouns.
1. agape (ἀγάπη, 26), the significance of which has been pointed out in connection with A, No. 1, is always rendered “love” in the RV where the KJV has “charity,” a rendering nowhere used in the RV; in Rom. 14:15, where the KJV has “charitably,” the RV, adhering to the translation of the noun, has “in love.”
Note: In the two statements in 1 John 4:8 and 16, “God is love,” both are used to enjoin the exercise of “love” on the part of believers. While the former introduces a declaration of the mode in which God’s love has been manifested (vv. 9, 10), the second introduces a statement of the identification of believers with God in character, and the issue at the Judgment Seat hereafter (v. 17), an identification represented ideally in the sentence “as He is, so are we in this world.”
2. philanthropia (φιλανθρωπία, 5363) denotes, lit., “love for man” (phileo and anthropos, “man”); hence, “kindness,” Acts 28:2, in Titus 3:4, “(His) love toward man.”¶ Cf. the adverb philanthropos, “humanely, kindly,” Acts 27:3.¶
Note: For philarguria, “love of money,” 1 Tim. 6:10, see MONEY (love of). For philadelphia, see BROTHER, Note (1).
