.. And do you think Luetjens would have given the permission to open fire at 5:52, IF the Hood and PoW would have been correctly identified earlier (5:45-5:50) ?
(My personal opinion is that he wouldn't... but that he SHOULD have... he was to warry and risked his ships to much...)
It would appear that Luetjens hoisted the signal to prepare to open fire (JD) at about the same time as Admiral Holland ran up his preparatory signal to do the same, thereby giving his tacit permission to fire. Thereafter it was up to the captains to fire when they were ready. I think the question was less identification than it was determining intent. If light cruisers, his worry would have been more about torpedoes than gunfire. Once Luetjens hoisted JD, Lindemann on the bridge and Brinkmann on Prinz Eugen were responsible for judging the proper times for their ships to fire and it appears that fire was opened the moment the range came down to the more or less optimum distance at which hits could be expected once target acquisition was achieved. Within moments of JD being hoisted, ID of the two new enemy ships as capital ships was made, on Bismarck at least, and the delay waiting for range to come down to around 22--23,000 yards was properly used to change the ammunition in the supply train (hoists, ring cars, etc., to APC, although the guns themselves were committed to firing the rounds already loaded into the chambers). With their limited amount of APC on hand (supply was divided between APC, base fuzed HE, and nose fuzed HE) Lindemann and Luetjens would have been conscious of the need to use the proper type of ammo and to wait until range was optimum for best results.
"If Bismarck was initially loaded with HE, wouldn't the easiest course of action be to fire the loaded guns and then reload with AP rather than unload the HE and then reload with AP?
A hit (unlikely, but one never knows) on the opening salvo from a 15" HE shell would still cause damage and, even with differing ballistics (If there were any) between HE and AP shells, data would have been obtained.
Bismarck was initially loaded with base fuzed HE, similar to common piercing or semi-armour piercing as used in the RN and USN. This was the ammunition type loaded in preparation for the attempted ambush on Wake-Walker's curisers. It would, as you suggest, be quicker and more expedient to fire off the rounds already loaded, and that was pretty much what was done on both German ships--Bismarck's initial salvos being base fuzed HE and Prinz Eugen's being nose fuzed HE. On Bismarck, once the guns were cleared of the base fuzed HE, proper APC was fired until the action ended. In Prinz Eugen, for unclear reasons, base fuzed HE was fired after the eight nose fuzed HE rounds were fired off.
As to the nature of the opening salvo, though often called a ranging salvo, these were more properly deflection salvos as guns with cold barrels will not shoot to the same ranges as later salvos when the barrels have heated up. Thus, the initial salvo is only loosely to find the range, but good for testing deflection.