The SOHC/DOHC thing definitely applies to the number of cams per head. Thanks to the W12 and W16's from VW/Labroghini/Bugatti counting the number of cams in the engine is no longer quite as practical.
Manufacturers started including the cam count in their engine descriptions as an indication of how powerful the engine might be. But now there are so many other additional items that indicate the relative power of the engine: number of valves per cylinder, variable valve timing, direct injection, turbo/super charging, etc.
Its actually more precise to refer to the number of valves per cylinder rather than the number of cams per engine. You can have 2 or 3 valve engines with SOHC, 4 and 5 valve engines with DOHC.
In our case we have a 24 valve V6 with variable valve timing.
-Old Army