The problem is in the English word "faith". There really is no direct Greek equivalent to the English word faith. The Greek word which is translated to the English word faith is πίστις [pistis]. Thayer's Greek dictionary has the following:
Thayer Definition:
1) conviction of the truth of anything, belief; in the NT of a conviction or belief respecting man’s relationship to God and divine things, generally with the included idea of trust and holy fervour born of faith and joined with it
1a) relating to God
1a1) the conviction that God exists and is the creator and ruler of all things, the provider and bestower of eternal salvation through Christ
1b) relating to Christ
1b1) a strong and welcome conviction or belief that Jesus is the Messiah, through whom we obtain eternal salvation in the kingdom of God
1c) the religious beliefs of Christians
1d) belief with the predominate idea of trust (or confidence) whether in God or in Christ, springing from faith in the same
2) fidelity, faithfulness
2a) the character of one who can be relied on
Part of Speech: noun feminine
It is a noun.
The verb form for faith in the Greek is πιστεύω [pisteuō]. Thayer's has the following:
Thayer Definition:
1) to think to be true, to be persuaded of, to credit, place confidence in
1a) of the thing believed
1a1) to credit, have confidence
1b) in a moral or religious reference
1b1) used in the NT of the conviction and trust to which a man is impelled by a certain inner and higher prerogative and law of soul
1b2) to trust in Jesus or God as able to aid either in obtaining or in doing something: saving faith
2) to entrust a thing to one, i.e. his fidelity
2a) to be intrusted with a thing
Part of Speech: verb
The Greek, just as with English, can be used in basically two different ways. (1) We can believe something or someone. That is to consider it to be true. Or (2) We can believe in something or in someone. That is, to not only believe it to be true, but to place trust in it. Sometimes when it is reasonably obvious that we mean the usage (2), i.e, to believe in, to believe it true and place trust in it, the word "in" isn't included. That is the case in Greek as well.
Thus, as in the English to "believe in", the Greek equivalent is "pisteuo en" or "pisteuo epi" or "pisteuo eis". And sometimes when the intended meaning "to believe in" is obvious the added words "en" "epi" or :eis" are absent.
The English is further confusing since "to believe in" is almost always used interchangeably with "to have faith in". When we say, "I believe in God" it means the same as "I have faith in God".
The real trouble begins, not so much the actual meaning of the words here, but rather how the belief, the faith, comes about. How we come to believe something or someone or how we come to believe in something or someone in every case except as it relates to God is well understood. We study the information, the data and sometimes our own experience associated with it and we either believe it or not. When we believe it to the extent that we place our trust in it, we say we believe in it. I think everyone, thinking rationally, would say that they believe in gravity. The information, data and experience with gravity is such that we would never doubt it. We always act in a manner never attempting to deny it of defy it.
I think the same thing to be true concerning belief, faith, in God. Some do not. Some think that, totally unlike anything else in life, belief, i.e., faith, in God does not come through information, data and experience but rather it is infused, injected, implanted miraculously. And that is where the difference lies in so much of the discussion about God.