I am not familiar with the k factor test and I see Terradyne have come a long way.
Please repost your photo... not found..
But "Some oscillation" indicates either stray RF or an impedance problem with fixture inductance.
Although a current source is high impedance the load I presume is just the current sense R and Vds at high ramp current. In other works a low impedance load measurement.
I would expect a purely resistive response unless bandwidth limitations are involved.
At low currents, the ringing suggests either the ramp is too fast OR some stray reactance using a high impedance low current. Now the circuit impedance is high,
Both stray capacitance and inductance affect the waveform.
If the oscillations also occur with a pulse load test, with ringing after shutoff, that would confirm this is a signal integrity issue.
Prevention depends on finding the cause and attention to controlled impedances and shielding.
I suspect k factor is useful for testing semi's on large ATE to avoid the layout bandwidth issues of a step pulse or an impulse as the ramp is the 2nd integral of the flat frequency spectrum.