Once you've decided who your narrator is, you decide what they should sound like. Anyone who insists that ALL narrators should speak alike is missing the boat.
You also need to consider when the narrator is telling the story. This gets pretty deep into the weeds of critical theory, but in short most novels are told in the past tense. Whether it is explicitly stated or not, this means that the narrator has had time to think about events and to put their thoughts into order. Essentially, the protagonist is writing a memoir of their adventure.
Thus, the narrator, while a character, should also be treated as being an author in their own right. They're not just putting down anything that occurs to them, or describing what they knew in the moment, they are massaging the story to tell a specific interpretation of events. From this point of view, the narrator is a character who is consciously editing what they tell you. So when the narration includes contractions it may appear natural, but it should be interpreted as the narrator having chosen to use that language for some specific purpose.
But in 3rd-person limited, it's an objective narrator.
Not necessarily. Most novels for the past 200 years use a style known as "free indirect discourse". This form of narration is so common that when it's explained in university classes many students have trouble grasping the concept: it sounds like the professor is just describing how fiction works. Basically, the idea is that most 3rd-person narratives are limited omniscient rather than a truly objective outside observer, and that means that the narrator has their own bias.
This writing style was first used in Jane Austin's Emma. In it, the narrator appears to be an outside observer, describing Emma in the third person, but as you read the novel you realise that the narrator is actually Emma's inner voice. By the time that you discover this, however, you've probably already accepted the narrator's descriptions of characters and events, so when things start falling apart the reader is as surprised as the character. It is only on a re-read that you really notice that the first thing the narrator told you was a lie, and that lie coloured your interpretation of everything that followed: "Emma Woodhouse, handsome, clever, and rich." -- she's actually spoiled, naive, and a poor judge of character.
My point is that since this style of writing is so ubiquitous, many authors, especially amateur and young authors, will use it without even realising that they are doing so. Thus, the narrator has an implied character that is somewhere between the protagonist and the author's own voice, but the author never actually planned out who the narrator is as a character.