Adding CC library PROCESS color to swatches palette in Illustrator not working

I have built 4 Creative Cloud Color Libraires, all colors are the "same", however built and analyzed differently:

PMS Spot Coated (built as spot)

PMS Spot Uncoated (built as spot)

CMYK Coated (built as process color)

CMYK Uncoated (built as process color)

When , I open a new document in InDesign / open my CC library and click on any color from ANY of the 4 palettes above, the swatch gets added to my "swatches panel" and names exactly as it is in the CC Library I built.

However, when I try to do this is Illustrator, only the Spot colors get added to the "swatches panel". The CMYK Process colors do not. If I manually "add" to the swatches, the name does not carry over from the CC Library. All of my colors have been built with a specific naming convention. I need the CMYK Process colors to be added to my swatches panel in Illustrator with the correct/same name that I built in the CC Library palette.

Can anyone help? I am assuming this is a bug, but hoping there may be a work around, that would not require me to rename manually. Please advise, of so! Thank you so much!