Would it make more sense if, instead of saying "left side of the screen", I substituted "left-aligned" for "left side"
Um...no. Unless you're suggesting that the right hand pane would be not left aligned.
, or if I said "old code to the left of the new code" and vice versa, or if I otherwise mentioned relative placement with respect to each other?
Yes, the general left to right thing makes sense. Because it's not referring to something that is false (at least in my case).