Weird problem with Acrobat and color space

I am working on an Illustrator file (AI). It has some raster scans imported into it, in AdobeRGB color space. 

The AI file has been set to AdobeRGB color space. 

I am saving copies of this file as a PDF.

One PDF I setup to be high res. In the Output stage, I choose:
Color Conversion: No Conversion

Profile Inclusion Policy: Include all Profiles

and export the PDF. I assume this will be in AdobeRGB colorspace (same as the AI file)

One PDF I make Low Res. 

In the Output stage, I choose:
Color Conversion: Convert to Destination

Destination: sRGB IEC61966-2.1

Profile Inclusion Policy: Include Destination Profiles

and export the PDF.

When I look at the PDFs in Acrobat side by side with the Illustrator file, they both look more saturated than the Illustrator file.

I open the PDFs and go to: Menu>Preferences>Color Management

Under Working Spaces it has:

RGB: Monitor RGB-sRGB IEC61966-2.1

For BOTH PDFs. The one saved out as AdobeRGB, and the one saved out as sRGB.

Is the Working Space the color space of the document? Or Working Space the color space that Acrobat is using, and converting all other color profiles to?

If its the former, something is very wrong.

If its the latter, then where do I find the colour space of the actual PDF file? I CANNOT find any other place dealing with color spaces.

NOW, one more step. 

For each of those PDF files, I Open them using Illustrator.

And when I check the document information BOTH show as being AdobeRGB color space. 

Despite one being AdobeRGB, and one being converted to sRGB.

Color is complicated, but this is doing my nut in.