Why are these measurements misleading and how do I fix them?
Since nobody has explicitly laid out what is probably happening here, I will attempt to do so. Most likely you are observing the effects of common grounding. The oscilloscope probe grounds are connected more or less directly to the earth ground of the AC mains power input to the oscilloscope. This connection is inside the oscilloscope and you can't disconnect it.

This means that both oscilloscope grounds MUST be connected to the same thing (because they are internally connected to each other). You can't connect one of them to Vb and the other one across Vr, because doing so effectively shorts those two terminals together (whichever two are on the oscilloscope ground).

There is also the possibility that Vb itself (or some other instrument or plug or cable you have not shown us) is internally earth grounded.

The reason the volt meter works is that volt meters, being battery powered and handheld, have no earth ground connection. Even mains powered benchtop voltmeters will probably work because their measurement terminals are generally isolated.

You don't necessarily have to get a differential probe. You can just decide where you want to ground your oscilloscope probes, and reference all channels to that same ground.

Everybody has to learn this at some point so don't feel bad.