There are several designations for most of the stars (left column in the picture). What is the name of designation for names like Alnilam, Sirius, Alhara ... How to call these names? Thanks
It's pretty simple: those are the names of the named stars, also called 'proper names'. All other designations you show in your table are catalogue numbers assigned to them.
While these names are historically grown, the official names are assigned / decided by the IAU to avoid confusion and standardize them.