I'm sorry to be delayed on this, but here at last is what I find the clearest response to our call for stories about armed heroes. The challenge was to find a modern instance in which a civilian bystander with a gun stopped a mass shooting.
@JamieBrowning submitted a May 26, 2008, news account from the Reno Gazette-Journal -- I was able to verify it in Nexis, though I don't have a link to the original -- of a man in Winnemucca who killed another man shooting into a crowded bar. As posted by Jamie Browning, the story contains this detail:
When Villagomez began reloading, a 48-year-old Reno man drew his pistol and killed Villagomez.
Note that the Nevada killer had to stop and reload, and that's when he was shot. So yes, an armed bystander saved the day, or night. And yes, his heroism happened when the killer paused. If so inclined, you could see this as an argument for the kind of gun measure that's now being proposed by Rep. Carolyn McCarthy (D-New York), which would take the fairly limited step of re-instating the ban on high-capacity magazines.
Thanks also to @Vic S and others for links to the 1966 University of Texas tower shooting and the 2005 Tyler County (Texas) shooting. Neither meets the strict criteria -- armed bystander stops mass shooting -- but each instance, armed bystanders do appear to have saved lives. Whether that's enough to justify calls for more guns as a way to stop shootings seems a little iffy to me, personally.