Because of satellite television, I would say classical - everyone will understand you, and although you will sound formal in your speech, you will also be able to read and write.
However, you will have difficulty understanding the dialect of the person you are speaking to. And if you have a dialect that you should understand, I would say it should be Gulf Arabic (Saudi Arabia, Kuwait, Qatar, Oman, UAE and of course my own Bahrain).
Economic (thriving free markets, not just oil-funded governments) and military activity in the Arab world is concentrated in the gulf, so unless you are trying to be a tourist (Egypt), the real value is in Gulf Arabic. Note that most Arabic teachers are Egyptian or from the Levant, and will try and persuade you otherwise ;)
I wanted to add my opinion as an Arab (from Bahrain), which is nowhere as scholarly as yours (I am so in awe of your linguistic nerdiness, thanks for such a lovely article).
Egyptian Arabic is dominant in a similar way to American English, i.e. Egypt had the largest movie industry and population, and everyone else could understand Egyptians while Egyptians had no need to understand other dialects.
What is interesting is the impact of satellite television. These new stations have global audiences and content that Arabs want to watch as opposed to their national censored channels. But their solution to pan-Arab broadcasting is classical Arabic, not Egyptian Arabic, and so there is a resurgence of the use of classical Arabic outside of courts and classrooms.
It's not funny, it's inevitable - Arab and Israeli tribes were all Semites (which is why accusing Arabs of being anti-Semitic is annoying, as it implies that we are not Semites) they share a common genetic and cultural heritage, and their languages evolved from common roots. If you ignore the current crazies on both sides, Judeo-Islamic tradition makes far more sense that Judeo-Christian one does.
A couple of corrections I feel I have to make as an Arab.
> There are the diacritics to think about, but outside of the Quran they are simply ignored.
They are not ignored - when they are present, attention is paid to them. I think what you mean is that Arabic speakers, knowing what the diacritics are, do not bother writing them down. That is not because they ignore them, it is because we have paid such close attention to them when learning Arabic that we no longer need to be reminded of them.
> That's still a big problem for a universal language of the internet though, since written Arabic is highly non-phonetic.
Arabic is highly phonetic, and if being phonetic was a criterion for being universal language of the internet, English should be disqualified immediately.
On arrival in England at the age of 10, I had no idea how English people knew how to pronounce their words. Now I know that non-Arab speakers may think the same way about Arabic because we do not write down the diacritics by default... but it is easy to buy books that have these diacritics written, and thus to crack the code.
But English seemed designed to trap foreigners into mispronunciations, to the great amusement of my classmates. (Traveling to America after college, it was mostly place names that tripped me up.)
However, you will have difficulty understanding the dialect of the person you are speaking to. And if you have a dialect that you should understand, I would say it should be Gulf Arabic (Saudi Arabia, Kuwait, Qatar, Oman, UAE and of course my own Bahrain).
Economic (thriving free markets, not just oil-funded governments) and military activity in the Arab world is concentrated in the gulf, so unless you are trying to be a tourist (Egypt), the real value is in Gulf Arabic. Note that most Arabic teachers are Egyptian or from the Levant, and will try and persuade you otherwise ;)