I wasn't really planning on getting into this discussion, in fact I was planning to keep out of discussions about SM's DW entirely, but. (Why does there always seems to be a 'but'?) IMO it's not so much what happened, it's how it happened. I don't think anyone is seriously trying to argue that Nine or Ten never killed anyone, considering that the fact that they did was a huge leitmotif during both their arcs.
But as I see it RTD tried to make a point of giving every death a certain weight, even if it was the death of the villain. Ten especially almost always had a moment where he found something beautiful even in the monsters trying to kill him. You were always aware that what ended was a life with all its possibilities and complexities, even if it was a life with a lot of wrong choices and few chances to get it right to begin with. Margaret Blaine was probably the best example of that, and going by The Writer's Tale he originally tried to put more of Davros's backstory in Journey's End, making the parallel between him and Ten, the war background in both their lives, more obvious. Almost every death was accompanied by the regret that things hadn't gone differently. Even with enemies like the Daleks, who don't have free will or individual thought, and no purpose but to hate and to kill, killing was never entirely off-hand or without consequences. Of course Nine and Ten killed. But they also both struggled with the consequences of that throughout their arcs. Nine in some ways recalls Stephen Bannerman from RTD's The Grand, coming back from the trenches of WW1, struggling with what he'd seen, the guilt of what he'd done, the guilt of having survived, and how killing had changed him; incapable of just picking up the life he'd known before the war, incapable of leaving the war behind. Dalek was all about the danger of becoming what you're fighting, and so was Boom Town and The Parting of Ways. Ten, and this, I think, comes out strongly especially towards the end of his arc, was so extremely distrustful of guns and violence because he didn't trust himself. He'd seen what the Time War had done to the Time Lords, how it perverted them so much that he had to kill them all along with the Daleks, and consequently he saw the seeds of that everywhere, especially within himself.
Sometimes there was no other solution, but it was very obvious that killing should not ever become something one should get used to or desensitised to. As far as I remember it was never portrayed as cool or stylish, or used as the butt of a joke, and I don't think that over the run of RTD's DW it was ever treated this... casually. If anything, RTD took it to the other extreme. The Doctor kept getting pitched against creatures like the Racnoss, the Carrionites, the Sycorax, the Daleks, or the Sontarans, who were determined to destroy or enslave the entire human race, had basically no individuality, very little complexity and very obviously absolutely no interest in the Doctor's offer. But even so, even then, there was a moment when you pitied them, when you were supposed to pity them, where you were supposed to realise that this might have been necessary, but was nothing to be applauded. It's probably no coincidence that even on the less family-orientated TW in CoE Jack, for whom killing had become something of a non-issue over all that time, is confronted with what exactly it means to take a life when it isn't someone nameless or faceless.
Which I think is a good thing. The older I get, the less fond I am of the casual, aestheticised violence we get in movies and on TV all the time, and the occasional break from that was nice.
What really got my attention in Day of the Moon wasn't even the way Eleven finally got rid of the Silence, because my interest in the episode, such as it was, had rather fizzled out by then; it was the complete lack of Eleven's reaction to Amy shooting the girl in the spacesuit. Leaving aside the problem that Amy had no way of knowing whether it was even the same person that killed the Doctor, which maybe isn't something that would occur to you unless you'd watched B5, the fact remains that the Doctor's companion shot a little girl in order to protect the Doctor, which I thought was a pretty dark moment. That she missed (or did she?) changes nothing about that, because she hadn't intended to miss. When she says later that she was glad she'd missed, this is still followed by, 'but you killed the Doctor', as if that was all the justification she needed. I can't even imagine what Ten would say to that, how he'd react. Rose in The Parting of Ways maybe came closest to doing something like that, but saving the Doctor is only part of her motivation, and is tied into a wider argument of trying to do the right thing. She ends up saving a lot more lives than his when she evaporates the Daleks, and even so it's still a complex enough moment.
Something else I found jarring was the sequence of River shooting the Silence; again maybe not even so much the fact that she did it, but how she did it; how the scene was filmed. Ten was genuinely shocked when Davros confronted him with the fact that he let his companions do the killing from him; Eleven just blithely delegates the shooting with barely a token protest. ('And I shouldn't like that. Kinda do, a bit.') The semi-sexual banter while talking about killing ('... but she'll definitely kill the first three of you.' - 'Oh, the first seven, easily.' - 'Seven? Really?' -'All right, eight for you, honey.' - 'Stop it.' - 'Make me.' - 'Maybe I will.') is really hard to reconcile with Nine or Ten. I'm not sure I would even have noticed something like that on another show, but in this case, coming from this character, I found such an overt sexualisation of violence rather off-putting.
On to the part that I have to admit I didn't actually pay a lot of attention to before it was pointed out to me. I've read the argument that the Doctor's actions merely amounted to a threat, as the Silence needed only to run and get off of Earth and no one would get hurt, but in the context of the scenes with the angry people in the bar and Nixon's guard aiming their guns at the Silence it didn't really look like that, and I certainly didn't get the impression that Eleven cared in the least whether or not the Silence would get killed or not. Thousands of generations, programmed to be mindless killers. (Or at least that's what he says. Is the programming passed on genetically? Or does everyone have to watch the recording of the moon landing?) And even if just one human being was made a killer against their will, without knowing about it, without being able to make a conscious decision about it or even remembering it, that's one person too many. Ten would have killed the Silence, if there were no other way. And angsted about it afterwards. But he wouldn't have turned the human race, or as much of it as watched the moon landing, into killers, or potential killers. The human psyche is complex and easily manipulated-planting a subconscious 'kill on sight' command into so many people's brains doesn't strike me as a particularly good idea.
There's of course the possibility that all this is intentional. One could argue that Eleven is a reaction to Ten's rejection of the gun Wilf offered, even at the price of a world full of Master clones, because he was so afraid of becoming something as dangerous as the Master himself, or even more dangerous. I'm also vaguely wondering if SM picked up the theme of the dark Doctor from Amy's Choice, who chose the image of old people with monsters hiding inside them for himself. But if it's that, shouldn't there at least be a bit more of an obvious hint? As it is, it's impossible to tell whether the Doctor of Day of the Moon is dark on purpose, or just accidentally dark because with SM clever plot twists trump consistent characterisation.