I always like the opprotunity to clarify

the "Baptist faith" (a convenient term that is OK because what we know what it means here) is actually the combination of core New Testament essentials with a thin veneer of Baptist distinctives.
"The faith" always supercedes any distinctives. We feel as though we best practice the commands of "the faith" in what is termed a "Baptist" structure.
There is but one church. Christ is the head. As local churches, we try our best to function as the church was doctrinally formed in the epistles. We need not the witness of history, nor a "building", nor a list of "succession" to be able to read the epistles and teach the truth.
Therefore, the question is moot in terms of the origin of "the faith".
You can trace the various groupings of Baptists to certain origins, but such beginnings are meaningless, since none of those groups pretend that its own formation is the "beginning" of "the faith"... they are merely beginnings of organizations built upon "the faith" that preceeded them and that will survive after them.