How do a generate a clean, precise pulse wave electronically without jitter?
recordings taken straight from audio out of my computer

The main problem with what you are doing appears to be expecting your computer to handle the DC content of your signals. It won't do this and it may well produce the unwanted artefacts you show in the pictures. An audio output or input on your computer is AC coupled; in other words it will neither record nor reproduce any DC content associated with your waveforms. After all, audio doesn't contain DC content so why should your computer (or any audio device) faithfully reproduce what you originally generated.

I would try and get hold of an oscilloscope and prove this to yourself.