I was trying to figure out exactly which tags LR Classic uses to store, in this example, facial recognition metadata. From what I can tell, and informed in part by the earlier replies, apparently LR does not store all metadata in tags (or sidecar files).
If the metadata actually were stored in documented (or at least discoverable) tag structures then there'd be some hope of being able to migrate that metadata to new tools. But since it's buried in the catalog in, from what I can tell, a completely opaque way there's no way I can see to migrate that data, manipulate it, etc.
I assumed that with the "Automatically write changes into XMP" option turned on, or with its manual equivalent *all* changes would be written. But I now believe I was mistaken.
In contrast, and for example, Picasa (optionally) stores its facial recognition metadata in well known and semi-standard tags.
LR stores everything shown in the Metadata panel, Keywords panel, and face recognition is stored in industry-standard metadata fields in the photo. The face recognition metadata is stored according to the Metadata Working Group's standard, e.g.
$ exiftool -a -G Photo_061505_002_edited-1-2-2.jpg | grep -i region
[XMP] Region Applied To Dimensions W : 640
[XMP] Region Applied To Dimensions H : 480
[XMP] Region Applied To Dimensions Unit: pixel
[XMP] Region Rotation : 0.16017, 0.00000
[XMP] Region Name : Peter
[XMP] Region Type : Face, Face
[XMP] Region Area H : 0.32505, 0.56974
[XMP] Region Area W : 0.24378, 0.40071
[XMP] Region Area X : 0.74400, 0.35106
[XMP] Region Area Y : 0.48796, 0.35815
Normally with "Automatically write changes into XMP" enabled, metadata changes are written back to the file nearly instantaneously, unless you've made large batch changes, which can take several seconds or even longer, depending on how many photos were changed.
If you're not seeing changes getting written back to the file, as a troubleshooting step, try saving manually with Metadata > Save Metadata To File.