South Sudanese are pretty different (hence the whole "two civil wars, succession from Sudan, and independence" arc of their history), but Sudanese from the North ("Sudan" proper, if you will), like you say, tend to identify as Arabs, though there are lots of different kinds of people in both places.
And then you have people like the Baggara Arabs who native to Chad, even further from the 'Arab' cultural center, and pretty much look like what most Westerners would say is a non-Arab black African, but they identify as Arabs and trace their lineage back to specific Arab tribes. Here's an example of one, photographed in Chad in 1910:
And then if you make Arabic-speaking a criteria, what do you do with the ethnic Kakwa people of Uganda,
some of whom speak Nubi, a Sudanese Arabic-based creole, and are descended from the Sudanese troops of Emin Pasha (1880s), who settled in Uganda during the British colonial period? This is the ethnic group that also includes people like Idi Amin, who nobody would mistake as an Arab.
Or, much more on point, what do you do with the Maltese? Maltese is basically a form of Arabic (rooted in what is called "Siculo-Arabic", the Arabic dialect once spoken on the Italian island of Sicily; even though it hasn't been spoken in centuries, there is a book-length grammatical description of it by Dionsius A. Agius), but very, very influenced by European languages like Italian, English, and French, to the point of losing all of its 'emphatic' consonants (ص ض ط ظ ع غ). There are additionally forms of Arabic that are native to Afghanistan (Balkh) and elsewhere in Central Asia, to Iran, to Turkey (Mardin and Hatay), to Cyprus (Cypriot Maronite Arabic, which might be the most divergent of Arabic dialects, being written in Greek and heavily influenced by it), and to Nigeria. Check out the book
Arabic as a Minority Language (ed. Jonathan Owens, published by Mouton de Gruyter in 2000) to see case studies of Arabic as it exists outside of the Arab countries, if you're interested. The language criteria actually makes this a very fascinating question, as you're immediately dragged far away from Arabia proper, even before Muhammad. The first ever Arab kingdom was founded in what is now Iraq by the Lakhmids, at al-Hira in about 300 AD.
They always have. Arabs are essentially nomads. Some academics even separate the Arabs from the Aramaeans in the pre-Islamic period in Arabia not based on language, but on whether they were nomadic (Arab) or sedentary (Aramaeans). Spencer J. Trimingham does this in his fascinating book
Christianity Among the Arabs in Pre-Islamic Times (Longman & Librarie Du Liban, 1979). There are historical/linguistic motivations to do this, as there are ancient peoples who are pretty solidly identified as Arabs who wrote in and presumably spoke Aramaic or Syriac, like the Nabateans. (See here John F. Healey "Were the Nabateans Arabs?" in Aram Periodical 1:1, 1989, 38-44.)
An alternate definition of an Arab? No. Again, I think if someone identifies as one, that ought to be enough. Conversely, if someone does not identify as one despite speaking Arabic natively and coming from an Arab-aligned country (e.g., Egypt, Syria, etc.), they ought not to be grouped together with the Arabs. Let everyone be whatever they are. Who really cares? The Islamic religion may privilege Arabness, but I am not a Muslim, and I think that's a really foolish mindset anyway. ¯\_(ツ)_/¯