The Greek words for faith "pistis" and believe "pisteuo" are two forms of the same word. "Pistis" is the noun form, "pisteuo" is the verb form. Nothing in the root meaning of either word carries any concept of works.
"pistis" actually has some range of meaning and like many words and phrases in the Bible
it takes some context and some harmonization of Scriptures to determine what it means and includes (the underlines are mine):
Bauer-Danker, Greek-English Lexicon of the NT (BDAG)
[BDAG] πίστις (pistis)
• πίστις, εως, ἡ (Hes., Hdt.+; ranging in meaning from subjective confidence to objective basis for confidence).
1. that which evokes trust and faith
a. the state of being someone in whom confidence can be placed, faithfulness, reliability, fidelity, commitment
b. a solemn promise to be faithful and loyal, assurance, oath, troth
c. a token offered as a guarantee of someth. promised, proof, pledge
2. state of believing on the basis of the reliability of the one trusted, trust, confidence, faith in the active sense=‘believing’
a. God: πίστις θεοῦ (cp. Jos., Ant. 17, 179.—Cp. π. καὶ φόβος ὁ τοῦ θεοῦ Theoph. Ant. 1, 7 [p. 72, 26]) faith, trust, confidence
b. Christ
α. of belief and trust in the Lord’s help in physical and spiritual distress; oft. in the synopt. gospels:
β. of faith in Christ, designated by the addition of certain words.
c. πίστις can also be characterized by an objective gen. of the thing: ἡ πίστις τοῦ ὀνόματος αὐτοῦ faith in his (Jesus’) name
d. πίστις is found mostly without an obj., faith, firm commitment
α. as true piety, genuine devotion (Sextus 7a and 7; ParJer 6:7), which for our lit. means being a Christian
β. Hb 11:1 defines πίστις as ἐλπιζομένων ὑπόστασις, πραγμάτων ἔλεγχος οὐ βλεπομένων. There is here no qu. about the mng. of π. as confidence or assurance (s. 2a above), but on its relation to ὑπόστασις as its predication s. under that word.—(Ps.-Aristot., De Mundo 6, 18 interprets πιστεύειν in someth. as incapability to see someth. that is apparent only to God.) Paul contrasts walking διὰ εἴδους e;ἶδος 3) as the lower degree, with διὰ πίστεως περιπατεῖν 2 Cor 5:7 (s. KDeissner, Pls. u. die Mystik seiner Zeit2 1921, 101ff). On the other hand πίστις is on a higher level than merely listening to Christian preaching Hb 4:2.
γ. πίστις abs., as a Christian virtue, is often coupled w. others of the same kind, esp. oft. w. ἀγάπη (agape):
δ. faith as fidelity to Christian teaching. This point of view calls for ἔργα (work) as well as the kind of πίστις that represents only one side of true piety:
ε. Ro 14:22 and 23 π. as freedom or strength in faith, conviction
ζ. In addition to the πίστις that every Christian possesses (s. 2dα above) Paul speaks of a special gift of faith that belongs to a select few 1 Cor 12:9. Here he understands π. as an unquestioning belief in God’s power to aid people with miracles, the faith that ‘moves mountains’
3.
that which is believed, body of faith/belief/teaching