I can't identify the headset because the part that typically has the label, the locknut, was missing. The upper headtube race and the adjustable race definitely don't match. They aren't even the same color, and they don't seat together properly. When I hold down the front brake and push the bike forward, I can see those 2 parts slide against each other as the fork moves. As far as the bearings go, it's anyone's guess what headset they're from.

I probably should have been paying more attention before I got the replacement locknut. It is pretty obvious the headset pieces already on it were hodgepodged together.

It looks like the whole lower assembly below the head tube is intact, but I'll probably just replace it all when I get a new headset.

If the Tange Levin sounds like a reasonable choice, I'll probably go with that.