Originally Posted by
genejockey
What it sounds like, to me, is nothing to do with the cassette or the chain. It sounds like your shifter is slipping.
With friction shifting, the screw that holds the shifter on also provides the tension to hold the shifter in place. The rear derailleur has a lot of spring tension pulling it towards the smaller cogs - that's how it works, after all: you pull the lever back, which pulls the cable which pulls the derailleur to larger cogs. Push the lever forward and the resulting slack in the cable allows the derailleur to shift to smaller cogs. The screw has to be tight enough that the spring tension doesn't pull the shifter out of place.
Possibly, and the chain being positioned on the second-smallest sprocket may be a coincidence, but there or on the smallest sprocket is where I expected to see the chain before the photos showed up.
If you're right, it's an easy fix. (Always check the easy stuff first!) All the OP has to do is follow your instructions.
And if the photos showed that the bike had Campy components from that era, I'd agree that a slipping shifter would be the first thing to check. I fixed plenty of those on my bikes as well as customers' bikes way back when.
But Shimano shifters from that era were generally good about staying where they were put. So, my guess: a 70% likelihood of worn chain and small sprockets.