The function generator settings affect only the function generator output amplitude at such low frequencies. You can consult your function generator manual, but below explains how it typically works.
When you set the function generator to High-Z you'll get almost the output amplitude you set with a 1K input Z. (Actually 1000/1050 = 0.952 x the set amplitude).
If you set the function generator to 50Ω you'll get almost double the output amplitude you set (Actually 2000/1050 = 1.90 x the set amplitude).
The function generator output amplifier has 50Ω in series and you are telling it what output load to expect when it determines the output voltage ahead of the 50Ω resistance.
So, aside from (perhaps) a much larger or slightly smaller than expected output amplitude there should be no great effect on the output waveform provided the function generator is not running out of headroom.
If you were to parallel a 1kΩ input impedance with 52.6Ω and set the function generator to 50Ω you'd get the set output amplitude (within whatever tolerances), but that's pretty much unnecessary and wasteful (and also limits the output voltage maximum to about half).
simulate this circuit – Schematic created using CircuitLab
As to why the designers of the laser driver chose 1kΩ as the input Z, it could be because they wanted minimum noise pickup if the input was to be left open or for some other design or system related reason.