Geerhardus Vos has some careful and precise thoughts about the various formulations in the New Testament (Reformed Dogmatics IV, pp. 80-81):
Usually, however, faith is presented as something that is directed to the Mediator, Christ. It is called “believing in Christ” (πιστεύειν εἰς Χριστόν)- actually, “into Christ” (cf. Rom 10:14). That is, the action of the soul by which it abandons its own doing and relies on the doing of Christ is presented as a local movement of the will into Christ. There is a relocation of the resting point of life. Where it formerly lay in the self-righteous sinner himself, it now comes to lie in Christ. It is also called “believing in Christ” (πιστεύειν ἐν Χριστῷ). The thought here is not so much of a movement into Christ as of its result, “resting in Christ” (Gal 3:26). Also occurring is πιστεύειν ἐπ’ αὐτῷ, “believing in Him” (Rom 9:33; 10:11, in a citation from the Septuagint of Isa 28:16), which has approximately the same meaning as εἰς Χριστόν. Finally, the apostle also speaks of a “faith of Jesus Christ” (πίστις Ἰησοῦ Χριστοῦ) with the objective genitive-thus a faith of which Christ as mediator is the object, a trust by which one depends on Him (Rom 3:22, 26; Gal 2:16; 3:22).
I think especially important here is the nuance of the formulation “believing into Christ.” It is a change of address. What so many Christians talk about so much is “Christ living in me,” or “asking Jesus into your heart.” The Bible does speak of Christ living in us, though never of “asking Jesus into your heart.” The Bible speaks a thousand times more often of us believing into Jesus Christ.