Shortly after 12 presidential candidates stepped off the debate stage earlier this month, roughly two-dozen influential Democrats at a watch party held an informal survey to see which White House contenders made an impression during the three-hour event.

The majority of the group—which included Los Angeles Mayor Eric Garcetti, who has yet to endorse a candidate, and big donors to former Vice President Joe Biden and Sens. Kamala Harris of California and Elizabeth Warren of Massachusetts—agreed one candidate was particularly...