There isn't a clear line. Indeed looking at it from a population perspective there really shouldn't be a line since genetically Europeans and Middle Easterners are the same cluster and anthropologically they are both Caucasoid. I mean sure there are distinct subtypes but if you focus on those you have to split up Europeans too if you want to be consistent.

I don't expect people to be familiar with 20th century physical anthropology or recent genetic studies though. Due to various cultural and political issues I expect at least some Middle Easterners would object to picking white/caucasian and many Europeans or North Americans would be surprised to know they are in the same category.

Generally I expect Turks and people from the Caucasus to pick white. The Asian parts of the former USSR really depends on who the person in question is, I'd expect them to pick White, East Asian or Other (as in mixed), not so much Middle Eastern. I'd expect at least a few Libyans or Yemenites picking "white" if there wasn't a Middle Eastern category, but quite a few would probably pick "other".

We could make a category to include both European and Middle Eastern since the cultural differences between them are captured by the religious & national background.