There are a few places in https://immersive-web.github.io/webxr/ that talk about "a responsible document" or "document is not responsible" with links to https://html.spec.whatwg.org/multipage/webappapis.html#responsible-document

That linked-to thing is a definition of a property of script settings object, and the property's name is "responsible document". There is no separate concept of "responsible" that applies to documents in the specs right now.

Maybe the spec means "is the responsible document of some settings object"? Does the spec really mean that, as opposed to, say "a document that has a browsing context" in the sense of https://html.spec.whatwg.org/multipage/browsers.html#concept-document-bc? (Those concepts are not identical, but they are for documents that are the active document of their browsing context, and doing the browsing context check would lead to a somewhat-natural fix to #872 as well, since you would then have a browsing context.)