How big was First Army?

How big was first Army? Figures from theBritish official history (James Edmonds, British Official History, vol.1.(London: McMillan, 1922), 42.) quoted German official accounts as: First Army, 320,000 and Second Army, 260,000. These figures are often quoted. They are handed down from one English language history to the next. It is common knowledge. Sort of an automatic factoid. It just does not check out with my limited knowledge. Germans eschewed using headcounts. Their preferred method was to list units. The British sources seem to favor nose counts. 

According to Volume 1 of the German official history (Reichsarchiv),First Army comprised 164 battalions, forty-one cavalry squadrons, 138 batteries with 796 guns and howitzers, and twenty-four engineer companies. Given that a battalion was roughly one thousand men strong in 1914, and given the relatively limited strength of the noncombatant rear echelons, it is clear the overall strength of First Army was unlikely to reach three hundred thousand at the start of hostilities. Further, the German official history (published in 1925) could not have sourced the British official history (published in 1922). According to the Sanitaetsbericht ueber das Deutsche Heer III Band page 36, the number was around 175,000. However, it was not clear which units this document counted; the number seemed low compared with our estimates that First and Second Army would have had about 250,000 men each. So, it is not undisputedly clear exactly how many men and equipment had to be squeezed through the Liège gap. But by any standards, German deployment would cover a large body of terrain. Why is it that in so many accounts the strength of HKK2 is not considered? Did they not also need to squeeze through? Something is fishy.