Which Death Star is in Star Wars Episode III Revenge of the Sith?

The dreaded legacy of the Galactic Empire, the armored space station known as the Death Star was a dark symbol of power used by the Emperor to oppress the galaxy. It is known that the Emperor commissioned the construction of least two Death Stars during his brief rule, both of which were destroyed by the Rebel Alliance; one in orbit around Yavin, the other in orbit above the forest moon of Endor. It is known that construction on at least one of these Death Stars began in 19BBY, following the end of the Clone Wars. It is commonly believed that this was the first, smaller Death Star that was destroyed in the Battle of Yavin (0BBY), with the second, much larger Death Star that was destroyed in the Battle of Endor allegedly taking only 4 years (0BBY-4ABY), following the destruction of the first, to be constructed to operational capacity (Construction on the Death Star in orbit above the forest moon of Endor may not have been complete but the space station was fully operational).

When considering the incomplete Death Star is estimated as being a third larger than its predecessor, possessed a more powerful Kyber Crystal powered super-laser with a dramatically shorter recharge time, and required a shield array situated upon the forest moon of which it orbited is it really plausible that the Empire could build such a large space station in such a short timeframe compared to lengthy construction time of it's forebearer. Is it not more feasible that both Death Stars were constructed in unison, with systems for the one in orbit above the forest moon of Endor being developed pending the success or failure of systems developed and tested on the smaller Death Star. 

This would mean that while the smaller Death Star only took 20 years to be constructed, after 24 years the larger Death Star was still incomplete; as evidenced in the opening dialogue of Episode VI: Return of the Jedi, whereby Darth Vader and Admiral Piett discuss the Emperors dissatisfaction that construction on the larger Death Star is not yet complete.

When considering the Emperors distrust of all of those around him, is it not possible that the Sith Lord foresaw Galen Erso's betrayal and Director Orson Krennics greed for power, and therefore assigned his most trusted naval officer, Grand Moff Wilhuff Tarkin, as commander of the smaller station, thus ensuring that his interpretation of the Sith Ultimate Weapon; the larger Death Star, was the most powerful it could be and his alone to control and use to spread fear across the galaxy.