How much background "noise" is normal on an oscilloscope
As an indicator that the second capture you displayed looks normal, here's one I just made of a 1 KHz square wave, from my ELAB-080 combination instrument (dual channel DSO, 16-channel logic analyzer, arbitrary waveform generator (AWG), and dual channel power supply) which also has a vertical 8-bit resolution in its DSO section.  (Note: at $495, the ELAB-080 was over your budget in your other question where I recommended the DSO-2090).

You can see the noise at the top and bottom of each square wave, which looks almost identical to your trace.