Okay, so during the Italo-Turkish war the Ottoman Empire put a lot of effort into using guerrilla warfare against the Italians, creating an entrenched resistance movement in Lybia that lasted so long that it fought against the Italians during World War 2, long after the Ottoman Empire had ceased to exist. However, when the Dey of Algeria beat up the French Ambassador to Algeria, the Ottoman Empire declared Algeria to be independent and let the French take it over. Likewise, when the French marched into Tunis, the Ottoman Empire just pretended that Tunis was an "autonomous region" of the Empire, and when the British invaded Egypt, the Ottoman Empire allowed looked the other way.
Why were these responses so different? The Ottoman Empire fought tooth and nail for Lybia, but completely ignored the loss of all of their other African territory. What made Lybia so special?
One quick clarification here: The Ottomans didn't 'fight to keep' Egypt because they never technically lost it; the British helped the Turco-Circassian ruling classes short-circuit a possible popular revolution and rising in the military under 'Urabi and his two confederates in 1881/2. (See Juan Cole's Colonialism and Revolution in the Middle East of 1993 for a classic treatment, which incorporates critiques of Scholch's 1980 German-original Egypt for the Egyptians!) Originally, the idea of having a Turco-Franco-British force do the work was put forward, but the French government actually fell over that question (among some others) and the Ottomans became very minor partners.
Until 1914, then, Egypt remained technically an Ottoman province, just as it had after the challenges of Muhammad Ali Pasha from 1805 to 1839. The Khedives held their spots on the throne and titular control over parts of the civil service, and the British de facto rulers were actually "Consul-General and Diplomatic Agent," not governors. This did not mean that the British didn't effectively run Egypt, or that everyone didn't know this (indeed it's often just colored red on period maps, or red with a strange special boundary on it). But the Ottomans still got their payments as well, and no longer had to worry so much about Egypt's recurrent financial problems. James Whidden's recent Egypt: British Colony and Imperial Capital (Manchester, 2017) gets at the oddity of Egypt's position, and its strange life as a colony-that-wasn't. The full story of how Egypt's existence shifted from Ottoman province to Mamluk governate, Khedivate, Anglo-French Dual Control, and then British stewardship is an interesting one; Cromer's own two-volume Modern Egypt (1908) is worth a read on this, and is freely available. He is a blow-hard as you'd expect of the Barings but if you can read past the Victorian bluster he is very informative. Vatikiotis's old final edition of Egypt: A Modern History hits the highlights of the entire era but misses a lot of the more recent studies.
As a postscript, after the First World War the British intended to take over Egypt directly. The Egyptians had other ideas, which forced the British to negotiate with the nationalists who had been trying to create meaningful organs of (at least upper-class) democracy since the first decade of the 20th century at least. The result was agreement on technical independence in 1922, although the British maintained key powers in later treaties and exercised brute force for example in 1942 to remove an Axis-sympathetic Prime Minister. That later control only truly dissolved with the Free Officers' revolution, the primacy of Nasser after 1952/3 shook out, and the Suez debacle.