Current is "measured" by Ohm's Law, as the load is a fixed 45R resistor, low inductance metal film. Changing from front panel to rear panel (with kelvin sensing) don't appreciably change the response.
Yes, this is (most likely) startup behavior. This particular initial overshoot is also observed in many other PSUs, especially newer one. For reference there's a thread on EEVBlog forum discussing this excursion:
https://www.eevblog.com/forum/testgear/lab-power-supply-turn-on-and-off-characteristics/
However I don't think this is caused by output capacitor discharge, the waveforms don't match. As indicated by the stairstep and correct slew rate, the upslope is within the PSU control loop and the PSU should be able to control its output. It just doesn't appear to control the current until a certain time after output is enabled.
I agree that a proper test setup will include limiting resistor in series with DUT, and if I want tight controlled source I'd use a SMU. It's just that it is kind of embarrassing that the PSU performs worse (w.r.t CC mode startup) than 5 decades old design using two opamp control with OR-ing diodes.