The map you link to is a basemap that is an ESRI basemap with contributions from Bellevue and describes Newcastle Way as SE 68th St.  Bellevue GIS shows the secondary arterial also as continuing all the way through to LK WA Blvd which is barricaded. While that may be its legal definition in their GIS or some transportation system plan, on the ground and for all users but city planners, zoning, and tax assessment, the map is probably correct. 

Based on those few iffy data points, I'm not sure I would make any changes here.