First of all: thank you all very much to have a look into this issue (if its really one)!
@vicentiu, its a DSOX1102A (https://literature.cdn.keysight.com/litweb/pdf/5992-3484EN.pdf?id=3019454) with about 1 MΩ ± 2% / 16 pF ±3 pF. The scope probes I have access to have the following specs:
Keysight N2140A (https://literature.cdn.keysight.com/litweb/pdf/N2140-92002.pdf?id=2803394) with 15pf - 30pf comp range - tested, undercompensated.
Probe Master 4902-2 (https://probemaster.com/4900-oscilloscope-probe-basic-kit-150-300-mhz/) with ~10pF - 50pF comp range (according to the page) - tested, undercompensated.
Rohde & Schwarz RT-ZP05 (500Mhz) - not tested yet. Should it be needed, I have also access to a higher end scope (1Ghz) and AWG.
I used the 1kHz square wave signal coming out of the WG (& test clips) of the Keysight scope as well as with Discovery Studios own generator. No luck with compensation.
My Analog Discovery Studio seems to be Revision D (sticker on the back, got it from Mouser).
I ordered the "original" scope probes @attila mentioned, they should be here (Switzerland) from a local distributor tomorrow. Btw: I'm not an electronics engineer (MSc in mathematics/computer science and sometimes no clue what I am doing), just a so called "Maker".