P: Include Subject Distance Metadata

"I think my custom metadata panel uses "com.adobe.subjectDistance".  Is that the same as Exif:SubjectDistance."

Yes.

"Is there any way I can access that data to display in my metadata panel?"

No, the Metadata panel won't display any of the proprietary fields in MakerNotes.  You'll have to use Any Filter (good for filtering and searching many photos at once) or the Metadata Viewer plugin to display all the metadata fields in one photo at a time.  Both plugins use the authoritative Exiftool utility to access fields within the photo file.

"Okay here's a test using an older picture with a definite distance of 3.16 meters."

That's very curious. The photo appears not to have the field EXIF:SubjectDistance and yet LR is displaying a value for it.

Could you please upload that photo and new sample photo that doesn't display Subject Distance to Dropbox, Google Drive, or similar and post the sharing link here (or send it to me in a private message)?  I'd like to determine precisely what's going on.

"It looks like EXIF:SubjectDistance doesn't return a value."

The Metadata panel displays what's in EXIF:SubjectDistance, so if that's not present, LR won't display anything. I'm not sure why older photos from the D200 would have that field but newer ones not -- perhaps a firmware update at some point?  

"I'm not sure of the accuracy of MakerNotes:FocusDistance.  I don't think the bird in the picture was 7.94 meters (26 feet) away from me.  It was more like 15 feet away."

I think the focus distance measured by the cameras is often inaccurate, which is why Adobe removed it from the Metadata panel way back in LR 3.4.1.   But I'm not expert in that issue...

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That indicates the camera has recorded in MakerNotes:FocusDistance (a non-standard, Nikon-proprietary field) 3.76 m in one photo and 7.94 m in the other:

Change MakerNotes:FocusDistance to EXIF:SubjectDistance and post a screenshot of that.

"When I said that my pictures before 2013 had subject distance and the pictures after that didn't I was
referring to the same Nikon D200. I don't think my D200 is causing the problem. I use a custom metadata panel that I set up in Lightroom. It includes subject distance. My D200 pictures taken before 2013 have subject distance displayed in Lightroom."

I only have four sample D200 photos (one from 2007?), and none of them contain EXIF:SubjectDistance.

You can see if the later photos contain the field EXIF:SubjectDistance using the Any Filter plugin's Filter command, as shown in the screenshots above.

Alternatively, upload one of those later photos to Dropbox, Google Drive, or similar and post the sharing link here, and I'll take a look at it.