I think that more clarification is needed in the question. The "position of the writer" is a bit on the vague side for me in terms of these three writers. Part of the challenge is that two of them, Conrad and Lawrence, are writing in the same time period and the other one, Rushdie, is writing in a vastly different one. To that extent, the position of the writer for Conrad/ Lawrence might hold some similarities to one another. In this paradigm, I think that the position of the writer was to explore the motivations and psychology of the individual. National and historical implications in plot development and story are secondary to the development of the context of the individual. In both of these writers, the individual's battle for moral and ethical understanding takes place in a situation that seems to be apart from cultural identity and context. The writer, in this setting, is meant to focus on the nature of human beings in their own right, away from contingency. I see Rushdie as being completely different. Emerging from the moment of Indian independence, when one nation became multiple ones, and living in England, one foot in one cultural setting, another foot in "the other," Rushdie sees the position of writer as still being able to examine motivations and ethical notions of truth, but having to do so in a cultural context and narrative that sees contingency as being very important. This is why the narrative voice in Rushdie's work is interrupted and imprecise, at times, because the cultural context of the protagonist is exactly the same. In Conrad or Lawrence, there is a sense of totality that allows the reader or character to fully make judgments and "objective" assessments of characters and events. Marlowe allows us to understand who Kurtz is and we grasp the judgment that needs to be made upon him. Lawrence is able to make transcendent claims about humanity and statements that are totalizing in effect. Both of them are able to provide a sense of "objectivity" because this is what they view as the writer's role. Yet, Rushdie strays from this. In part due to the fact that culture, as a concept, is far from defined, far from totalizing. Rushdie's characters are not able to really exact a distinct and final judgment. They are, like culture is, ambiguous, vague, and imprecise. In Rushdie's setting, the writer's position has to be one of reflection of contingency and context, elements that define who we are and what we do.
I must say that this clubbing of authors is quite odd. The three you have mentioned are not only vastly divergent in terms of their aesthetic credo, but also products of completely different historical and cultural situations. Although Lawrence and Conrad are generally seen as Modernists, strictly speaking Conrad is a late-Victorian in whom we see a definite anticipations of many Modernist tenets and D.H. Lawrence is more aptly in historical terms, a participant of the movement of literary Modernism along with Joyce, Eliot and Woolf. Salman Rushdie is an acclaimed postmodernist author on the other hand. He is more into narrative experiments and language-games and all that.
Conrad's authorial position is often seen in complicity with the imperialist project, primarily because of his portrayal of the Africans in novels like Lord Jim and The Heart of Darkness. There is an alleged construction of the Africans as cultral others and a retelling of the civilizing myth of the colonial project.
Lawrence is more of a libidinally committed author, all set to oppose the social hypocrisy and double-standards regarding the idea of sexuality especially. He has faith in the life of pure senses and focalizes on the mythologizing, primitivist and instinctualist powers of sexuality. The Oedipal myths in his novels, the animal poems, the poems about love both maternal and erotic and so on, Lawrence is always set to speak the libidinal truth in a daring psychoanalytic way.
Rushdie is a multiculturalist pastiche-maker, a witty re-worker of cliches and someone who uses a multilingual register in his politically potent fictional texts of magic realism. As a cultural and political exile, he is concerned with relocating the homeland of the dispossessed diasporic subject in terms of crafting an 'imaginary homeland' through the work of art.