The same voltage on both sides of the jumper means its resistance is very low. That's what you want and rules out a problem with it. But make sure it's a solid connection that doesn't change when moved around.

I can't imagine a voltage drop of just 1 V to be enough to prevent the starter from engaging, but who knows? Is there any sound when you turn the key and the starter fails to engage?

You could do some further investigation before spending your money.
Since starting from the starter terminal works fine, I would measure the voltage on that point with respect to ground (negative battery terminal) and repeatedly try to start the engine using the ignition switch.
If there's no voltage or a very low one when the starter fails to engage, I'd suspect the ignition switch. Perhaps wiggling the key (not the carpet) helps? If so, definitely the iginition switch.

Linked below is a good document to have when trying to find electrical faults in a 200-series.
www.v8volvo.se/mekartips/volvo/Filer%20240/Elsystem.pdf