The camera doesn't DICTATE your photography, no. How can it? It's just a machine. However, my images looks subtly different from camera to camera. Images from my medium format camera have a different viewpoint because it has a waist level finder. This is most obvious when shooting portraits. Having said this, I would choose a certain camera knowing I have a certain job to do. I'd not choose to shoot medium format if I needed to work fast for example.
Then the OP assumes that we all use DSLRs and all shoot exclusively digital.
If I was discussing the differences between DSLRs, then I don't think it makes any difference whatsoever if you are using a D810 or a 5D MkIII... absolutely none. All DSLRs essentially work in the same way. Some may be faster than others, some may have better low light ability etc, but you use them in the same way, so the images will be the same.
Choice of medium - film or digital - that has a huge effect on the outcome.
It's an obvious question I'm surprised has been asked. The answer is obviously, yes, and no, and is certainly not the same for everyone.
Everybody says the same thing "oh my camera is just a tool. Th camera doesn't make you a better photographer" but for me, I am a bit more honest and I have to say that I need to have an enjoyable camera on my hands in order to be inspired to photograph. I am sorry if somebody doesn't agree with me , but I need to actually like what I have in my hands. It doesn't need to be a Leica , but it doesn't need to be a 20£ point and shoot. I just need something that doesn't compromise my photographs.
Not agreeing with you is not what bothers me here, its that you assume anyone who DOES disagree with you is being dishonest. Don't you think that's a wee bit arrogant?
I genuinely don't give a stuff about cameras. In many ways, I hate the ****ing things. They promote pointless gear based rivalry and boring conversations amongst amateurs about what lens is better than what other lens, and we end up with images all looking the same - taken for the same reason: Because people are inspired by cameras and not the subject. Amateurs treat cameras like jewellery or status symbols in the main. A modern codpiece..... I have a 5D MkIII with L series "glass" so I am powerful. Look at me.
I understand about not having something that compromises your photographs... of course.. if it did, then it's the wrong tool for the job, but to need a certain kind of camera to be inspired to take a photograph means your interest in photography is the cameras... the actual mechanical and optical processes. For me, it's the subject. I don't care what I use so long as it's the right tool for the job. The camera certainly will not be inspiring me. This is why some photographers don't even take the photo themselves. Rankin for instance, or Gregory Crewdson... they have someone else operate the camera. A great many amateurs will inevitably now say "Then it's not his photograph". Why? Because he never pressed the button? Is that all photography is? Pressing a button? They do so as to not be distracted by the part that requires no skill... pressing the damned button. They've set up the shot, realised their vision, lit it, arranged it... whatever needs to be done is done... pressing the button can be done by any fool. Ridley Scott never actually operated the cameras either, but he's still the director of the film - it's still HIS film... not the camera operator's.