ACR renders the raw NOT Lightroom! Therefore, the statement: Lightroom does not use the Camera Raw plug-in, isn’t clear or accurate. One of them has to render the raw, it’s ACR that is doing the rendering as Victoria clearly points out. Again, that’s why LR pop’s a warning if ACR is an older version than LR in terms of processing. ACR can’t render the raw with the new features of LR, LR suggests you use Export (or forego the newer processing).
There is no suggestion made to use Export, that I can recall. Lightroom's external editing workflow is quite unrelated to Export, both conceptually and practically.
IIRC, LR offers either "Open Anyway" or "Render in LR" in cases where there isn't full ACR compatibility. The latter option switches to the same pre-saved file method, that LR employs with a third-party editor; also, that LR employs with as external editing preset even if ACR is compatible. In all these cases, ACR is beside the point.
If you install LR on a computer that simply lacks a compatible or near-compatible PS version, I don't think you see any warning message at all - LR goes ahead and hard-renders the image to a file made in accordance with the current external editing settings (filetype, bitdepth, colourspace). The external editor executable may also need to have first been set up manually.
One immediate way to tell the difference, is that in the ACR-mediated workflow the document title seen inside PS reports the source image's filename (for lack of anything better) - this might be a Raw filename such as "IMG12345.NEF". At this point there is not yet any edit image showing in LR. If when the file is FIRST saved you pick a different filename or location or even file format, instead of the one suggested for you, LR will still track and import that. If you don't save, nothing is imported.
OTOH in the LR-rendered workflow you will see a PS document window titled e.g. "IMG12345-edit.TIF". That's because it is already saved. And a new image version is already present inside LR, corresponding to this, before you have done anything in PS. If you don't then save from PS, or if you save to another file instead, you simply lose those changes so far as LR is concerned - but the file IS already imported within LR regardless.
The other change in the case of a non-compatible older ACR (or no ACR present, and/or no PS present): LR greys-out the Edit In PS.. As Layers / Photomerge / HDR / As Smart Object choices in its menus. Those workflow options are only offered when LR knows that it could - if required - instruct PS to open the image / images concerned (via ACR).
AFAICT these workflows are all about LR "scripting" PS operations behind the scenes. But IMO the only sense in which Lightroom is "using" ACR, is the sense in which when I get my hair cut, I am "using" some scissors: I am using them by proxy, though the involvement of the barber
