The headline is a tad hyperbolic. Notably, there was one area where Lawrence proooobably was, uh, stretching the truth (for good reason): he insisted that the Arabs had sneaked some ninja division into taking Damascus "first." This account isn't shared by anyone else and would imply these Arabs took the city, then retreated, then came back later or something.

Why would Lawrence fudge here? Well, this will seem very quaint to modern eyes, but apparently there was a bias toward "whoever did the work gets the spoils" back then. If the British army took Damascus then it was theirs to distribute as they saw fit, if it was a combined British/Arab force then it'd be a little different. This is ludicrous since there's a unified commander-in-chief who is British who can pick whichever army they like to have the honor of entering the city first. Anyway, Lawrence was exaggerating the date where the Arab troops arrived to try and get them some of the glory & responsibility afterward, since he was considerably more pro-Arab independence than some of the other Brits.

It's a silly issue since the Turks had basically lost regardless and who cares which army enters first, of course.