Darth Lefty The pin you drew through is just the mast lock. The steering axis is through the front yellow head tube.
The weird motions are due in part to the hinges which loosen a little over time and need re-tightening, and the mast hinge, which is the hollow tube below the saddle bag below the water bottle. I have used a tapped a large awl in at both sides which seems to tighten that up for a while as well. The longish bolt and nut I use to fasten the top of the steering mast actually threads into the chrome mast and holds the mast tighter than the original part. Also the slight flexing of the long steel seat mast adds drama, but not as much as the long one-piece aluminum seatpost on my 20" folder. That has a newer one-piece steering mast which has very minimal deflection when properly tightened. You can see my previous posts on the yellow Getaway in the folding bikes section by googling "bikeforums bobbyg getaway"

@Smaug1 I appreciate your concern for my hands, but padding my bars goes back 20 years to my early 40s when I developed tender "crampy" hands. I think some of it had to do with an over-zealous prescription of statins from a former doctor. My hands are a little better now, but still get a little tender now matter which of my six bikes I ride. Occasionally I try to adjust and ride one of the bikes without the padding, but more than 20 minutes is all I can take. Also batting and catching softballs aggravates my hands. I have, however been able to reduce the padding on my seats in the last 7 years.

Also, on this bike I tend to ride with my palms resting on the hinge of the brake levers which are not padded and are very comfortable. (Google "bikeforums BobbyG crampy hands").