Here are a couple more suggestions to drill in students' heads that sequence means progression and series means sum.
Consider a sequence of "episodes", each of which is a set of events. Episodes can be added by taking their union. The events that have happened so far as of an episode is the sum of it and all previous episodes. We call this sum a TV series. If your students aren't ready to accept a union as a type of sum, you could start with an "anyone can die" series"anyone can die" series, define an episode's value as the number of deaths so farin that episode, which gives the series as the total body count.
Consider a sequence of "ballgames", each of which has a value of 1 for a win or 0 for a loss. Then the World Series is a sum that approaches 4 for one of the teams.