This may have been addressed previously in this long thread, but in case not ... maybe this will help alleviate some confusion over the number of Woodward's cavalry unit.
Confederate Kentucky units were formed in several different locations, and sometimes records did not get to Richmond to assign "official" numbers to some of these units. Also, some units started as battalions but were later increased to regimental strength, but didn't get (or want) a new number.
Then ... after the war ... some of these units were given arbitrary numbers to try to keep them straight.

What this means for Woodward's cavalry unit is, it was called when formed, the 2nd Kentucky Cavalry. However, there was already a 2nd Kentucky Cavalry under John Hunt Morgan and Basil Duke (Morgan's original regiment). These two units never served together, so there continued to be two "2nd Kentucky Cavalry CSA" regiments throughout the war.

After the war, when counties started writing to the War Dept. to verify service for veterans pension applications, the clerks in the War Dept., in sorting the original muster rolls and other records out, assigned arbitrary numbers to some units to differentiate them from others. This happened to the units called 1st Ky. Cav., 2nd Ky. Cav., 3rd Ky. Cav., and 7th Ky. Cav., all of which were duplicated during the war. Woodward's unit was arbitrarily called the 15th Kentucky Cavalry, but this is purely an artificial number. There was never any 15th Ky. Cav. CSA during the war, and Woodward's unit was never called the 15th until long after the war.

Personally, I wish the War Dept. clerks had not done this sort of thing decades after the war, because it causes a lot of confusion today.

How the "3rd Ky. Cav." gets into these descriptions of Woodward's unit, I don't know. That was a separate unit entirely.

Hope this helps!