"We have song sessions; you have tune sessions. They are both sessions."

I'd venture to suggest that by "session", most of us understand that to mean a 'tune session' as you put it. Even Snuffy himself (in the Upton thread) says :

"Always lots of fringe at Upton, John - just about every pub has sessions or singing. Come and join us. " - sessions OR singing. Snuffy - may well join you for a tune (or a song) at Upton :-)
M