Is there a good term to differentiate between these two modes of play? I've heard both of them described as "solo adventuring."
One Player Only, One Character Only - like it says. One guy with one paper man. No assistance from player-run PCs.
One Player Only, Multiple Characters - I'd throw one PC, many assistant NPCs in here - a fighter with his warband, a mage with apprentices, a hot-shot fighter pilot with his wingman, etc.
Back in the day, we only used "solo" for the first one. If you brought NPCs with you, it was just "one player" but not "solo." Sometimes I feel like I need a term that makes it clear which one is which.