Although the capabilities, features and results from Lightroom's Develop module and the use of ACR in Photoshop are identical, the default place where the editing information is stored is different.
Lightroom by default stores the edits in its catalog file. ACR stores the edits in an XMP file. Neither is destructive.
Lightroom never modifies the pixels in your original RAW file, so in answer to your question, you cannot save/ export this modified RAW from Lightroom.
I realize that layers and adjustment layers can only be saved from PS as a PSD (and maybe TIFF) format. But what if I did something like a spot heal or clone stamp to touch up photo, is there then a way to save the adjusted/new pixels as part of the raw file? Or is the only option at that point to save the psd file?
Seems like I have to save as jpeg and "rename" and then I have to re-import to get the jpeg files back into LR.
This defeats the purpose of Lightroom. Any time you want the edited photo to appear in Lightroom, you don't have to anything. By default, Lightroom shows you the edited photos. There's absolutely no need to "save" the adjusted/new pixels somehow just so you can see the edits in Lightroom. (And also, let's stop talking about doing a "save" from Lightroom, as there is no such function).
If you want the adjusted/new pixels for use OUTSIDE of Lightroom, then you do a File->Export and choose a JPG or TIF. Again, as I stated, there is no need for this exported photo to be re-imported back into Lightroom, as Lightroom will show you the edited photo whenever you want, by default, you don't have to somehow "turn this on".
There is a setting in the Catalog Settings, View > Catalog Settings, to Auto Write changes to XMP. If that option isn't checked then the edits only get stored in the LR catalog file. With that option checked they get saved to both the sidecar XMP and the catalog file.
PS does not Edit RAW image data. When sending a file to PS, even though it may appear with the extension of the RAW image, it starts to create either a TIFF or PSD file depending on the option you have selected in the External Editing tab of the LR Preferences.
So when you Save that image in PS the extension will turn to either TIF or PSD.
If you want to do more ACR types of edits in PS to an image there is a Filter for that. Filter menu item and then Camera RAW Filter. That opens ACR to make those edits.
Anytime you use ACR from Bridge or PS a XMP sidecar file is created because neither PS or Bridge has a Database file to store the edit in like LR does, IE the LR Catalog file (It is a database file).
No. RAW files are never changed in either LR or ACR. This is Non Destructive editing. The edits get saved to a XMP or the catalog with LR and ADDED to the image when viewing. The only way to Bake in, Burn In, the edits is to either save it as a TIFF, PSD or JPG image in PS or Export it as one of those formats in LR.
Just open your RAW file from Lightroom as a "Smart Object in Photoshop and you can use ACR in photoshop to adjust any setting you made in Lightroom. This is assuming both programs are using the same Camera Raw version. These edits DO NOT transfer back to the Lightroom adjusted RAW although there is a way to do this with LINKED FILES in Photoshop but it is a little more complex and you have to keep track of making sure the changes are saved and read from the XMP files.
Thanks, both your and Just Shoot Me, are correct, but apparently can have only one correct answer