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Strong (4103) gives 67 occurrences in the New Testament of the word πιστός, pistos, which is mostly translated 'faithful'.
The relationship of the Greek word πιστός, pistos, to the Greek word πίστις, pistis (usually translated 'faith' or 'faithfulness') see Strong 4102 is the same as the relationship of the English word 'faithful' to the English words 'faith' or 'faithfulness' ; that is to say adjective to noun.
The word 'believer' is somewhat weaker than the word 'faithful' in English as the word 'believer' does not carry the same connotations of behaviour and character as does the word 'faithful', see below for the Oxford English Dictionary definitions. 'Believer' denotes an adherence, or a membership, rather than being a description of characteristics.
This meaning is that which is now the usage of the word in common parlance, it should be noted.
Also, using an adjectival description,'faithful', is more true to the original Greek than is translating πιστός, pistos, (an adjective) with the word 'believer' (a noun).
The word 'believer' does not occur at all in the KJV (see Young's Analytical Concordance) but the word 'believe' occurs multiple times, translating πιστεύω, pisteuw See Strong 4100
Theology. A person who believes in a god or the doctrines of a religion; esp. a Christian. Sometimes: spec. a member, esp. a fellow member, of a particular Christian group.
Oxford English Dictionary - 'believer' 1.
b. Of friendship, love, etc.: loyal, constant, steadfast.