I am looking for a word which describes the talking an artist does between songs in a live performance. The word "intermission" is close, but I think that it has the connotation of a longer break, possibly with the artist doing nothing (on stage). Another word that comes to mind is "patter", but this word can also mean something like "glib talking", rather than the talk between songs.
A very standard term is stage banter.
An example of usage:
Stage banter––what you say to the audience between your songs––can make or break a live performance, and it's something that's especially important for artists who are just getting started.
—B. Baur, Say What?! Upgrading Your Stage Banter, Music Connection Magazine, July 30, 2017.
The term I've heard used for this between-songs talk is "stage patter." Instances of the term go back to the early twentieth century in a Google Books search. For example, from Israel Zangwill, "The Serio-Comic Governess," in Collier's Weekly (January 3, 1903), we encounter a character who says:
And the tales you tell me—how useful they'll come in for stage-patter!
The term seems originally to have referred to a carefully constructed series of remarks—as much scripted as the rest of the stage show. For example, in "The Point of View," in Scribner's Magazine (August, 1920):
Isn't much of the slang of this generation manufactured in the word factory of some writer of musical comedy or stage patter instead of in the home or the village shop where our grandfathers' slang—and furniture—was made?
But more recently, the term has come to refer specifically to the type of half-scripted/half-extemporaneous between-music spoken interludes you seem to have in mind. Thus, for example, from an unidentified article in in Frets, volume 6 (1984) [combined snippets]:
Try out a little rehearsed "stage patter," to get an idea of what it's like to talk to an audience. It usually goes a long way toward relaxing everyone (yourself included), if it isn't overdone. There's a fine line between keeping the set moving forward and rushing through it. A bit of humor or a few words of introduction here and there within a set can keep the audience keyed up for what's coming next.
And from J. Randy Taraborrelli, Motown: Hot Wax, City Cool & Solid Gold (1986) [snippet window no longer active]:
"Maurice King was just the best," says Mickey Stevenson. "He worked with Dinah Washington, Billie Holliday, and all of the big shots of the forties and fifties. As well as teaching the Motown groups their music, he taught them about stage patter, what to say in between songs. Diane [Ross] would work with him day and night on this."
And from Alyn Shipton, Groovin' High: The Life of Dizzy Gillespie (1999):
Like all seasoned troupers, his stage patter included an element of the well rehearsed alongside the spontaneous wit.
Patter can be used generally for any interstitial talk by an artist during a live performance. Patter also has a more specific meaning: the talk done to keep the audience from getting restless while the artist is preparing for the next song -- for example, while adjusting a capo on a guitar.
However, if the artist is specifically introducing each song, I would suggest using the same word radio DJs use for that purpose: intro. If they are telling the audience about what they just heard, it would be an outro.
Note that on recordings where spoken word has been spliced between tracks after the fact, the word "patter" should not be used as there is no audience to keep momentarily distracted. (Such segments are sometimes referred to as "interstitial material").