In the Landnamabok, Svalbard seems to be Jan Mayen, so it was discovered and named by the Icelanders some 500 years before the Dutch.
Why has the name 'Svalbard' been transferred to Spitsbergen which, from the Landnamabok sailing directions, could not possibly have been Spitsbergen?
Hi - short answer from Rolf. It has been a debate for many decades, mainly following the Spitsbergen Treaty signed in 1920. The use of 'Svalbard' was mainly favoured by Norwegian historians and those who preferred this name for political reasons rather than historical ones, to make a statement of the archipelago being Norse/Norwegian rather than Dutch or anything else. What 'Svalbard' originally referred to is uncertain, Jan Mayen is one option that has been suggested, together with the east Greenland coast, Spitsbergen or the pack ice edge. (what makes you so certain that the 1194 entry in the Icelandic Landnámabok referred to Jan Mayen?)
Around 1920, the term Svalbard was hardly used. The treaty as signed in 1920 did not have the term Svalbard at all anywhere in the text. Nevertheless, it is today commonly referred to as the 'Svalbard treaty' although it should be the 'Spitsbergen treaty'. I guess you should direct your question to Norwegian historians and authorities to check why this is so.