The case of saving Sirius is fundamentally different from going back in time to save dead people. Simply put, Sirius was fully alive and well at the moment that Harry and Hermione initiated the time travel. They were not trying to change the past; they were trying to change the future. The issue they were dealing with was not that Sirius had been wrongfully killed, but that he was about to be wrongfully killed. (Well, technically have his soul sucked out, but for the purposes of this answer we can simplify things.)
An important principle of time travel in the Harry Potter books is that whatever the characters do while back in time already occurred the first time around. This is most obvious – and indeed the point where Harry understands – in the case of the patronus. In the original timeline Harry was saved from the dementors by someone across the lake casting a patronus. At the time he did not realize who it was, but when he was in the past he realized that it was himself, and so he cast the patronus as his future self. That was not a change of the past, but a fulfillment of the past.
Given that this is how time travel works, the past cannot really be changed. If a character's future self would go back in time to save someone, that person would never have died in the first place. When someone dies in the original timeline that is a proof that no future characters will go back in time to save them.
We can now try to understand what the purpose of going back in time in Prisoner of Azkaban was. After all, if the past cannot be changed why would someone ever go back in time? The answer to that is, as we started with, that they were not planning on changing the past; they were planning on preventing the future. Nothing that had already occurred stood in contradiction to Sirius escaping. The only problem that they were facing was that in their current situation they did not have any means to save Sirius. Going back in time was not meant to change the past; it was meant to give them the ability to do what they would be unable to do as their present selves. Knowing the events of the three hour period enabled Harry and Hermione to position themselves so that they would be able to undetectedly help Sirius.
If you look carefully at Dumbledore's words, this becomes even more clear:
"What we need," said Dumbledore slowly, and his light blue eyes moved form Harry to Hermione, "is more time."
Note the second to last word. Dumbledore did not say that they need to change the past, or do any number of strange things to time. All he said was that they need more time. Sirius was about to have his soul sucked out, and there was no time for them to do anything to stop it. Going back in time was simply the means of gaining extra time to conduct the escape plan.
That all covers Sirius. With Buckbeak it is a bit of a different story. The events involving Buckbeak were certainly in the past at that point. Whatever had happened at his execution had already occurred three hours prior to the initiation of the time travel. As such, it should have been impossible for them to change anything. And indeed it was impossible. They did not change anything; they merely caused the fulfillment of events as they had initially transpired with the influence of future characters. That is to say that Buckbeak had never been killed, because Future Harry and Future Hermione had saved him.
However, as mentioned above, given that the past cannot be changed, what caused Buckbeak to be saved in the first place? There would be no motivation for Harry and Hermione to go back in time to save him, since he had already been saved. While it is true that Harry and Hermione did not actually know that Buckbeak had survived, Dumbledore was aware of this, so he shouldn't have had any motivation to send them back in time to save Buckbeak.
We can address this with what I alluded to in this answer. Bucbeak's escape was only incidental. If Dumbledore had not needed to help Sirius then he would not have sent Harry and Hermione back in time, and Buckbeak would have in fact been killed. Once they were going back in time Buckbeak became a necessary accessory for Sirius's escape, and therefore had to be saved. Because he had to be saved during their time traveling, he ended up being saved all along. When Dumbledore first saw that Buckbeak had escaped he was presumably confused. But once he realized that they needed to go back in time to save Sirius, it was obvious that the best way to save him would be by using Buckbeak. He thus figured out that it had been his plan all along that allowed Buckbeak to escape, and therefore hinted to Harry and Hermione to use Buckbeak in that manner.
So to conclude, no past events were changed in the act of saving Sirius. A future event was prevented, and the means of preventing the future event was what caused the past to happen the way it did the first time around. Time-Turners could not be used to change any other events in the past because that would exceed the scope of a Time-Turner.