Hi,
i´d use the second one with a voltage ref at the pot input.
And a capacitor at V+ input to avoid noise - especially during the pot adjust.
With this solution you always need a current meter to adjust. If you need some fixed values like: 10mV, 20mV, 50mV... i recommend to use a rotary switch to select them.
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Maybe an advantage.
if you have a power supply with 5V for example you could use a 40:1 voltage divider at opamp output. As feed back use the output of the voltage divider.
Benefit that i expect is less noise, lower zero level (maybe 1mV). and additionally you could use a comparator with 4.5V to OPAMP output to detect overload.
It all depends on the used opamp.
Klaus