Why did Snape tell Bellatrix about Dumbledore knowing he was returning to the Dark Lord?

Question

Snape: By allowing Dumbledore to think that I was only returning to the Dark Lord’s side because I was ordered to,

_{Harry Potter and the Half-Blood Prince (2005)}

I am probably being a bit naive, as I'm only reading the series again for the first time in a good few years, but why would Snape say the above to Bellatrix, rather then just not mentioning it? As now, to Bellatrix/Voldemort, this looks like Snape has informed Dumbledore, and Dumbledore is aware Snape is returning to the Death Eaters. Yet, if Snape is and wants to continue being a spy for the Death Eaters, from the perspective of Voldemort/Bellatrix, surely this would compromise Snape's and Dumbledore's connection and would mean Dumbledore would stop trusting and giving him info, and ultimately fire him.

So in simple terms, from the perspective in which Snape is speaking to Bellatrix in, Dumbledore is aware Snape -- through Snape telling him -- is now a Death Eater again. Surely this would lead to Dumbledore not trusting or confiding in Snape, which surely Voldemort would do something about, as what would Snape now be doing, considering, in theory, Dumbledore wouldn't be giving him information and he wouldn't have a job there?... Then after, how does Snape continue on the pretence he is a spy for the Death Eaters, if based on what he has told them, Dumbledore is aware he is working with them.

Therefore, that would mean he wouldn't be trusted and create the loop of the fact he wouldn't get any information, most likely not work there due to being fired or leaving and would not be able to be a spy. So how doesn't Voldemort/Bellatrix pick up on this when questioning Snape and consequently believe/accept his pretence of being a spy for the Death Eaters on the Order?

P.S.: it probably isn't simple terms. Also, this is all from the perspective of the Death Eaters, based on what Snape's told them, as we all know in reality Snape is a double agent for the Order.

Answer 1

1. Dumbledore orders Snape to rejoin Voldemort, in order to act as a double agent for the Order. He recognises that Snape will declare that he is loyal to Voldemort.

2. Snape tells Bellatrix (and Voldemort) that he is loyal to the Death Eaters and acting as a double agent for them; he explains that he can get away to meet with them because Dumbledore believes that he (Snape) is acting on behalf of the order.

Both sides believe Snape is working for them, while pretending to act for the other side.

As FreeMan points out in the comments, this also (out of universe) creates ambiguity for the reader by keeping it unclear where his true loyalties sit.

Answer 2

The key point here is that Bellatrix has outright asked him, among other things, why he took so long to go back when summoned, and Snape is carefully answering all her questions. By answering in this manner, and stating that the Dark Lord is satisfied with his answers, he has confirmed to Bellatrix that Voldemort is convinced that he is working for the Death Eaters against Dumbledore, and that Dumbledore does not suspect his actual affiliation. And of course Voldemort cannot be fooled.

Answer 3

In short, Dumbledore ordering Snape to return provides him with a plausible reason for returning when and how he did and more importantly, it only requires that he deceive Voldemort in his intent, rather than his actions.

Snape has been ordered to return to the Death Eaters and does so willingly, but at the same time, he not only has to convince the other Death Eaters that he is sincere, he has to convince Voldemort himself. The issue is that Voldemort is an accomplished Legilemens (Order of the Phoenix, Ch. 24), so while he can't exactly read minds, he is very skilled at telling when people are lying to him and he can learn some things from people's thoughts.

Part of Snape's success in doing this is that he is a highly skilled Occlumens, able to shield his mind against Legilimency. However, if he returned to Voldemort with his mind completely occluded, he could never be trusted. He needs to be as honest and open as possible, at least with Voldemort, and having a compelling, honest reason for being there with the bare minimum of lies or omissions is the smartest, safest path. Snape is nothing if not intelligent and careful.

Once he has successfully convinced Voldemort using a true version of events, it only makes sense that he would give the same explanation to the other Death Eaters. It would be highly suspicious if he had told them one thing and the Dark Lord something else entirely.

The following is an excerpt from Order of the Phoenix, Chapter 24: Occlumency, in which Snape is explaining Legilimency as part of attempting to teach Harry Occlumency.

'Only Muggles talk of "mind-reading". The mind is not a book, to be opened at will and examined at leisure. Thoughts are not etched on the inside of skulls, to be perused by any invader, the mind is a complex and many-layered thing, Potter — or at least, most minds are.' He smirked. 'It is true, however, that those who have mastered Legilimency are able, under certain conditions, to delve into the minds of their victims and to interpret their findings correctly. The Dark Lord, for instance, almost always knows when somebody is lying to him. Only those skilled at Occlumency are able to shut down those feelings and memories that contradict the lie, and so can utter falsehoods in his presence without detection.' - Order of the Phoenix Ch. 24