Yes, in the literal sense, Willy (Walter Slezak)'s part is "the second refrain." But verse can also be used in cases like "To Keep My Love Alive" to refer to its 4 "verses" (continuations of the main song), rather than "I've been married and married, and often I've sighed..." (technically, the verse in the song). Still, a musicologist might well call them refrains. I while back (though I can't remember who the singer was), there was an album of just the verses of Rodgers and Hart songs (e.g. "Though I know that we meet every night" for "With a Song In My Heart").
How funny that you should mention "I Feel at Home With You" and "Cherry Pies Ought to Be You." Right now, I'm working on a song with Alex Rybeck and when he asked for prototypes, I cited those very same lyrics.