Here is a link to the actual morning states of the British on various days in the winter of 1854-5 (ref), and here is the data in a nice modern table (click to embiggen), with the same numbers processed into the Federal US equivalents:
Note the British have more strength categories, and if we'd had some different tables they'd also separate out those Present Under Arms (PUA) into those detailed to the trenches and those available for other duties. We might also separate those "on command" (roughly "extra duty" in US terms), Bâtmen and other employees (roughly "special duty" and "daily duty").
The strength PUA is those available to fight, rather than on other duties. When we read strengths in many British Napoleonic and other histories (like Oman's magnum opus) the "effective strength present" is the total of officers and men PUA, excluding men "on command", sick etc. - see here for an example of Sir John Moore's Army of 1808 (ref). This leads to a distortion, because French strengths are generally reported as roughly "present for duty" and a distortion in favour of British fighting prowess occurs similar to US-CS distortions.
So what of Smith's claim that the British only had 8,000 troops actually present, ignoring the fact that he called the 1st (Guards) Brigade, 1st Division the "Guards Division" and completely missed out the largest division in the army (Light Division)? Simple - he's ignorant of the fact that these numbers, which are perfectly valid and derive ultimately from Kingslake Vol. 6 (ref) are only that portion of the army actually engaged (excluding the Light Division).
Anyone who'd read any regimental histories (or Kingslake) would know that Inkerman was fought by those portions of regiments not assigned to the trenches. Take the 88th Foot (The Connaught Rangers) for example marched to Inkerman with the Grenadier, No. 5, No. 7 and Light companies, and were joined late in the battle by No. 2 company after they'd been relieved by No. 3 company (ref); No.s 1, 4, 6 and 8 companies were on trench duty. The four companies that set out mustered only about 290 men, but this implies 725 effectives in the battalion as a whole. There were 31.2 battalions of infantry (exc/ Marines) with the army and the 3rd November morning state shows 18,498 infantry officers and men PUA for an average of 593 - the 88th may have been a strong battalion, although it is not clear the returns above include the detachment at Balaklava.
As one should note, the total present increased even though the combat strength stayed static due to illness. There was a steady flow of replacements to the old regiments, but 12 full battalions (the equivalent of a full infantry Corps de Armee in 1862 terms) reinforced during the period*.
Here we can see the British had far more sensible measures of strength, but stulti caveant for those who don't understand it!
* Arrival dates:
12th Nov - 62nd from Malta
20th Nov - 97th from Greece
27th Nov - 9th from Malta
4th Dec - 90th from England
9th Dec - 34th from Corfu
15th Dec - 17th from Gibraltar
19th Dec - 89th from Gibraltar
22nd Dec - det. 71st from Corfu
26th Dec - 18th from England
1st Jan - 39th from Corfu
19th Jan - 14th from Malta
3rd Feb - remainder 71st