The least they can do is either number or codename the shots. How do I ask the local pharmacy if they have the updated shot? "The one they've been working on for a while?" "Just came out?"
Call it booster shot #, give it a more interesting name, or call it an amalgamation of the strains it targets.
Something like CV19B-MDRN-202309 would be nice. But as we're moving into the maintenance phase of this virus, I imagine that it'll just come to be known as the "Fall 2023 COVID shot", similar to flu shots.