Well, I'm happy to say the problem with the Triumph appears to simply be a dead battery. I discovered this when I went out to try and get it running, and noticed that the headlights were pretty dim.
Yay for the scientific method.
Anyway, I charged up the battery and the bike started right up. However, the "Check Engine" light is on. In the interest of not ending up twenty miles from home with a dead battery, I rode the Vespa again today. Tonight I'll put my voltage meter on the battery and see what happens.
The Triumph still has the original battery, so it's about due for a new one. This weekend I'll have to go and get a new one.