13 R3, which win10 image to download?

I can, and that is indeed strange.  The "display names" of the main Windows 10 versions have always been Home and Pro, never Professional -- although the internal edition codes for those versions are actually Core and Professional.  If no additional details are shown when you select them, then depending on the format of the file you get (I've never used that tool), if you were to download both, you might be able to analyze the Windows image contained within each to learn more about them before choosing one, but I would probably just go with the one that has the correct name and larger size, the latter of which could indicate a newer version of Windows 10 and might therefore save you some time updating.  The alternative would be getting a standard Windows 10 image directly from Microsoft, but that would not come with the drivers and other applications that would have shipped with your system.  The latter could be good or bad, depending on your preferences.  Some people like to start with as clean a slate as possible and install what they actually want even if it takes more time to get to a fully working system, whereas others want their system fully functional as quickly as possible and want the extras like Alienware backgrounds and such.