This is sort of an analytical question. Having seen Rogue One, I re-watched some of the scenes from A New Hope, including the two scenes above and analyzing the dialogue, the answer seems to be "no." Tarkin once says he believes that it is time to test the station's full destructive power, not its destructive power. Note: I didn't have the whole film on me, so I might have missed some key lines.
A New Hope does specifically state that Alderaan was the first planet destroyed by the Death Star, as mentioned by Valorum.
Tarkin: Not after we demonstrate the power of this station. In a way, you have determined the choice of the planet that'll be destroyed first. Since you are reluctant to provide us with the location of the Rebel base, I have chosen to test this station's destructive power... on your home planet of Alderaan.
However, there is, as far as I can tell, no indication that it hadn't fired before. Tarkin mentions that he wants to demonstrate the "full power" of the station.
Tarkin: I think it is time we demonstrate the full power of this station.
Rogue One does not contradict this, as the Death Star
only destroys the city of Jedha and the Scarif Imperial base, never an entire planet or moon.
The Rogue One Ultimate Visual Guide reaffirms this.
It is only a fractional test: a small portion of its total reactor yield is siphoned into superlaser blast targeted at Jedha City, The results are unmistakably devastating, and a humbling preview of the power a full-powered test could produce.
A New Hope also never implies that Alderaan is the Death Star's first target, only that it's the first public one.
Tarkin: You're far too trusting. Dantooine is too remote to make an effective demonstration. But don't worry. We will deal with your Rebel friends soon enough.
In Rogue One, Tarkin mentions that the Death Star is still being kept secret from the Senate.
Overall, there seems to be no implication in A New Hope that the Death Star had never fired before.
Having seen Episode IV many times, I always assumed that what was shown in the movie was the first super-laser test. The particularly relevant lines are:
Until this battle station is fully operational, we are vulnerable.
This implies that some functions of the station are already functional. But that goes without saying. Life support obviously works, since they are having a meeting onboard.
The final checkout is complete. All systems are operational. What course shall we set?
In each case, there is no specific claim that the weapons systems are not operational to any extent. However, that seems to be the implication.
Moreover, in Episode VI, it is much more strongly implied that the second DEATH STAR's weapon is either fully operation or not functional at all. Lando is shocked that the battle station is capable of even destroying a rebel cruiser.
That blast came from the DEATH STAR! That thing's operational!
I does not seem like an intermediate level of super-laser functionality was countenanced in the original trilogy.
Remember that Rogue One takes place in the weeks or days before ANH starts. So news of what really happened to Jedah City and Scarif would not have had time to disseminate. Furthermore,
Palpatine informs the Senate that Jedah City was destroyed by a mining accident. Scarif, being an Imperial base, would likely not have had anyone outside the Empire notice the attack at all in a timely manner
What Rogue One does is turn up the need to demonstrate the Death Star so that people would quickly become afraid of the Empire.