In the beadarray package we used to refer to the numeric identifier as the ProbeID, which IIRC is the Array_Address_ID in Illumina's nomenclature. The IDs of the form ILMN_1234 were referred to as Illumina IDs in the beadarray documentation, but I guess the official manifest calls them Probe_ID.
Ultimately it doesn't really matter what the name is, as long as you're aware that the CodesBinData column holds some sort of identifier and you are confident you can map that back to a genomic locus correctly.
Great! Thank you for your quick and informative comment.
I can go further with your answer.