Is the (primary) user permitted to use National Instruments software for private purposes under the second-use right of his commercial license?

No, private use is prohibited under the NI License Terms. In principle, the named user or main user of the license may install a copy of the software on his or her private computer. However, the latter may then be used only for work carried out within the framework of his employment relationship and not for projects of a private nature.

Source: Software License Agreement_National Instruments as is 2017-01-17