I have seem similar crosstalk issues with cheap logic analyzers using standard fly leads. That was solved by replacing the fly leads with coaxial leads. If you have connected the ground lead for each channel, then I would not expect crosstalk due to the cable. If you check the pin-out of the 40-pin pod connector (see
Keysight Probing Solutions for Logic Analyzers datasheet page 71/72), every even wire is connected to a ground connection (assuming you connected the ground connection for each channel). So any crosstalk would happen in the connectors or adapter boards (which look like double sided boards without ground planes). I doubt that the edge rate is high enough for crosstalk to happen in those few centimeters. The fact that the impedance of the wire is different probably will not make that much difference at this length, unless you are pushing the limits of the pods. I believe these pods were used with logic analyzers up to 800 MS/s.
I suspect that this may be ground bounce, especially since you are using a solderless breadboard. If you search for the term than I am sure you can find a more detailed explanation, but in short ground bounce happens when the impedance of the ground lead is too high compared to the currents that are flowing to this. It happens, for example, if you have an IC with two dozen signal pins switching at very high speed but only one ground pin. The current of all those signal pins will have to flow back through the ground pin. This current flowing through the inductance can induce a voltage drop, raising the effective ground potential for that device. The way to solve this is to lower the impedance to ground. This is why fast parts, especially those with many pins, will tend to have many ground pins. And why the pod allows for one ground connection per channel, even though HP/Agilent/Keysight specifies that you may only need to use one ground wire every four channels or so if your edge rates are sufficiently low (so more inductance can be tolerated).
The single, probably not very low inductance/resistance, ground connection on your solderless breadboard may be responsible. Try lowering the inductance of the ground leads, for example by directly connecting them to the demo board, or at least connect more wires from demo board to the ground strip on the breadboard, one after every two or four wires to the logic analyzer pod.
The original passive pods were designed to work with much longer cable, but they were twisted pair, hence a well-defined impedance. I do not believe the woven cables are resistive, only the leads plugged into the pods are. I do not own an HP MSO, but from the pictures it looks like a 40-pin 100 mil connector on the scope end. That means that the cables for the 16500-series logic analyzers will not fit. See
this auction for pictures of that cable. And even within that series there were multiple models matching different logic analyzers with differences in the housing of the connector. I believe the original cables+pods for the MSOs are quite rare. It is not unusual to find a MSO + leads for less than the leads alone.