Your observation about white privilege is profound; I'd argue that it's all the more reason why "White" *should* be capitalized. Although your ancestry comes mainly from Poland and Lithuania (from what you describe), you still have a common Eurocentric experience in America that's similar to those of people with English, Irish, or Scandinavian backgrounds.
Your question about Spain and Italy is a good one. Ultimately, it depends on what other ethnicities are mixed into one's genealogy. Italy is a part of Europe. So if someone with Italian heritage has multiple other ethnicities from Western Europe and Northern European mixed in with their Italian heritage, it's fair to classify them as White. Especially if they have very light skin, like I do; I'm part-Italian, but I also have German, French, Swedish, Dutch, Serbian, and Austrian ancestry mixed in -- which accounts for my relatively pale complexion, even though I'm part-Italian. There are Italians of Color, however. Someone could have a ton of Italian heritage but also have Middle Eastern, Northern African, or Central Asian ancestry blended into their genes. They most likely appear olive or tan in skin tone; that individual would be an Italian of Color (or, in America, a biracial/multiracial person), in my view....particularly if they have a tangible familial connection to their Arabic and/or Asian heritage.
The same concept would apply to people whose families come from Spain. There are White Spaniards as well as Spaniards of Color. Multiracial actress Charisma Carpenter is a perfect example; she has both Spanish and Cherokee ancestry alongside of the parts of her familial background that are European in nature. If you see her on the street, her skin appears a bit darker than we'd envision "White people" as being, even though it may not be quite the traditional hue of medium-brown in skin tone that most people imagine when they think of Latina women. You could even look at a country like Portugal; approximately 95% of Lusos are "White" (due to its proximity to France and other Western European countries, and their resulting ethnic mixes) but there are some Lusos of Color as well...largely due to Iberian, Northern African, and South American intermarriages.
The Middle East is another case. Much of the procreation within the Middle East over the centuries has been a result of migration from East Asia, Central Asia, and North Africa. But, obviously, there've also been European influences in Middle Eastern family trees. This is why there's such a heterogeneous spectrum of skin colors in a nation like Turkey -- it's truly a "gateway" between multiple continents. It depends on the experience of an Arabic or Middle Eastern person, whether they consider themselves "White" or "BIPOC." Why do so many Persians and Arabs get racially profiled when in airports and stores? Even though the U.S. Census Bureau still considers people of Middle Eastern background to be "White," that is largely based on white supremacy and the precursors to "replacement theory" -- that, if Americans with Arabic & Middle Eastern heritage are allowed to identify as BIPOC, then that would reduce the percentage of "White" Americans on paper (which would be antithetical to the goals of white supremacy). It's utterly racist that the MENA community doesn't have an applicable category with the option to check on Census forms...especially since many MENA groups have requested one!
This ties into Jewish identity. Mike Shouhed from the reality show "Shahs of Sunset" is arguably BIPOC (do a Google image search for him!). Yet, he is Jewish...not Muslim, the way most of his castmates are. Most of his immediate family tree comes from Israel. Yet, they are still Brown. He has undoubtedly been racially profiled throughout his life while living in Los Angeles. National identity in and of itself doesn't determine "whiteness." Contrast that with what you've described as your Hebrew heritage. Even if you have distant ancestors from Israel -- how much intermixing exists in your family tree, over the centuries, from your ancestors who were Eastern European? (or Western European?) The Shouhed family doesn't necessarily have that genealogy. And what about someone in America whose family is half-European, half-MENA/Arabic? Yes, light skin tone contributes to privilege...but it isn't always so clear-cut, because there's always going to be SOMEONE in America with darker or lighter skin pigmentation than someone else's. That's the difference between White Jews (those with a bulk of their ancestry being Western European and Eastern European, especially the latter) versus Jews of Color (where a critical mass of their recent ancestors were from countries southeast of Cappadocia).
So it really is more complex than "The Paper Bag Test." Charisma Carpenter would probably be derided as "too White" by many Americans with very dark skin tones; conversely, she'd probably be called a "mulatto" (or some other racist pejorative) by white supremacists. People who fall in the middle range of the color spectrum tend to get ignored, dismissed, or slip through the cracks...not everyone is necessarily pale-as-fuck like little old me, or a deep mahogany in their complexion like Danai Gurira or Morris Chestnut.
