Dave
I'm guessing that the screenshots you have shown here are from a version of Photoshop CS, rather than Photoshop Elements which Matthew is using. An assumption based on one of your screenshots showing the option to open the image as a Smart Object, which PSE does not support.
I have been looking at this with my copy of PSE 9, which might or might not be the same version as Matthew's, but is unlikely to be much different and I can see the same issue he describes.
So far as I can tell, PSE simply defaults to 240 DPI when opening Raw files. There is an option in the image editor preferences to set the default DPI for new documents, but ACR blithely ignores this. For a while, Adobe has been simplifying Elements, presumably to make life easier for those users who do not always understand the complexities of image processing. I am certain the function was there in PSE 6, another casualty being the ability to use printer profiles when printing, which also vanished somewhere between versions 6 and 9. Know what you are doing and want more options? Hey, just spend a chunk of your monthly disposable income on Photoshop CC instead.
The question is, does this matter? I have long considered the DPI setting to be more documentary rather than having any real effect. It is simply telling the output device how large the image should be displayed. It is irrelevant for monitors, which default to their native resolution (usually 72 DPI) when showing images full size. It does have a bearing on the size of prints where there is a direct correlation with DPI. A lower DPI reduces resolution, but allows a large print to be made and vice versa for higher DPIs.
Here, Adobe has been clever and in the PSE print dialogue it is possible to set both the print size and DPI, the latter being independent of the DPI setting for the image. What happens if the two are not the same? I don't know for certain, but I suspect that Adobe resizes the image in the background, most probably using bicubic interpolation which works well. You could do the same manually using the Image Resize function but why bother when it can be done automatically? Mind you, why print in PSE at all when Lightroom has the same options to set image size and DPI independently and so much more when printing? Not only that, but It has better Raw conversion and comprehensive library management to boot.
Sometimes, such as when submitting images to magazines for example, there is a requirement for a specific DPI and for this PSE still has the option to change the DPI setting. Adobe is actually providing everything which is needed, but in a way which is different to more conventional methods.