Travis picking descended via Mother Maybelle, Merle Travis and Chet Atkins as a form where the guitarist plays the bass rhythm, picks out the melody (in the bass or the treble depending on style and mood) and fleshes out the chord accompaniment with however many fingers and talent he has left. It can suit an accompaniment, as a stand alone instrumental performance or a tasteful solo.
Folk style finger picking, or finger style perhaps, descended from Folk guitar, probably includes a form of alternate bass, and is primarily intended as an alternative to strummed rhythm guitar, as a tasteful accompaniment to a volcalist or some other style of soloist.
With the proviso that many guitar players are an informal, come as you are collection of musicians often self or group taught, with a varied (including zero) interest in music history and theory and for some whom jazz is just an extra set of chords in the blues, while any form of finger-style is also finger picking, or travis picking, or whatever else they want to call it that day. And popular music is very much an oral and loose tradition that is lived more than it is studied.
Of course nothing is ever easy and while all travis picking is finger style, not all finger style or finger picking is travis picking, but not that you would notice, unless you had made a point of listening to a lot of travis picking and read a bit to know just what the difference is, or should be.
I don't see Maybelle Carter included in that descendency. It's well documented that Merle's greatest influence was Mose Rager, with Ike Everly also having some influence. Before Mose was Kennedy Jones and before him Arnold Schwartz. These early players were very much in the blues genre. Big Bill Broonzy's playing wasn't a million miles from Merle's style. My take on it is that the finger picking of some blues players was simply as an accompaniment to their songs and that developed into the folk style of playing whereas other players like the Rev. Gary Davis and Blind Blake predominantly featured virtuoso guitar picking. It was the latter approach that Arnold Swartz and Kennedy Jones embraced, leading eventually to Merle's style. Merle is quoted as saying that his style of playing should be more accurately called Rager Picking.
First let's dissect the folk style of finger picking. It is in its fundamentals, pattern picking. The pattern can be varied thus, T, 1, T, 2, T, 1, T, 2. It can be reversed thus: T, 2, T, 1, T, 2, T, 1. Thumb (T) plays strings 6, 4, 5, 4, or 6. 4, 6, 4, depending on the chord, or chord voicing.
Travis Picking is more free form, less dependent on patterns, The thumb takes care of bass lines, and in Travis' case, the index finger picks out the melody, and any fills used between vocal phrases. The heel of the picking hand damps the bass strings and is rested somewhere near the bridge.
The style has been modernized by players such as Chet Atkins, Tommy Emmanuel, and Richard Smith who make use of two, three, or maybe even all four fingers of the picking hand.
I haven't mentioned Jerry Reed, because he took the style to a whole different place, almost creating a separate style of finger picking. I hope this clears up some of the mystery.
My first guitar teacher was a finger picker. He taught me the basics about what he called Route Picking. However I really wasn’t interested in acoustic guitar in high school so I quit after a few months.