Nice bike! In your last picture of the headset, you are missing the black plastic top-cap. Is that because you removed it for the picture or because it is missing? If it’s missing that would explain a lot of movement as it creates something for the clamshell to press against when it closes.

As as for a tool, I just use a Robertson screwdriver and a hammer to tap it. Just put the end of the screwdriver into one of the notches and tap the end of the screwdriver until the lock ring comes loose, then unscrew by hand. Keep taking it apart until you reach both sets of bearings so that you can regrease. That should eliminate some play as well. Anytime I’ve felt “looseness” in the handlebars, the headset has been the culprit.