The boundary used in the "Handbook of the Birds of Europe, the Middle East, and North Africa: The Birds of the Western Palearctic" is the most accepted. Furthermore, I understand this is followed by the most popular Western Palearctic checklist, i.e. the Association of European Rarities Committees (AERC) list.
The boundary is set at the 27th parallel so Kuwait for example is in but Bahrain is out.
For my purposes I can fly easily to 4 northern Saudi cities. Tabuk, Hail and Arar are well above the 27th parallel but Jubail is exactly on the line. If I bird in the north of the city my observations count but if I bird in the south of the city, they don’t!
avadavat near Riyadh
There are other researchers who put the boundary further south. Some argue for all Arabia to be included. Most argue for all but the south and south west. Either way there is no general acceptance of their arguments for the greater boundaries. I am comfortable to work within the boundaries in the handbook.
common myna near Riyadh
Bird Tour operators market Kuwait as the extreme south eastern edge of the Western Palearctic. This is a bit of a stretch! The northern suburbs of Jubail in Saudi Arabia work out to be the recognised south east extremity!
white throated kingfisher near Riyadh
The recognised boundary leaves Riyadh and the rest of the country in the Afro-tropical region.
By the way you may have noticed the three bird pictures of species that breed near Riyadh. What they all have in common is that they are not Afro-tropical or western palearctic but essentially Indo-Malay. That's another story.