You write that you believe
writing in the first person gets the reader closer to the character. As far as I know, this is generally accepted as true.
This is wrong.
What brings a reader closer to a character is narrative perspective, not grammatical person. Third person narration is the standard in Western fiction. Readers are used to this convention and do not perceive it as signifying an outside perspective.
Narrative perspective has nothing to do with grammar. You bring a reader "in line" with the experience of a character by narrating that experience in a manner that evokes a similar experience in the reader. This can be done in both first and third person.
First person, as Mark Baker has elaborated, may come with a more subjective perspective and include misperceptions, lies, and other deviations from the objectivity that we commonly associate with the "third person omniscient narrator", but there are third person narrators that lie to the reader and first person narrators that manage an objective view on "themselves". As you note, I put themselves in quotation marks, because we must never confuse the narrator with the character. The narrator is a literary device and it – not he or she – is always distinct from the protagonist, as well as the author, no matter how much the narrator takes on one or the other perspective.
As a writer, you must keep three concepts apart:
- the author
- the protagonist (or any other character)
- the narrator
The narrator is the device with which you guide the experience of the reader. The narrator has several dimensions, of which the following are a few:
- interiority vs exteriority
- grammatical person
- subjectivity (and lies) vs objectivity
- showing vs telling
- present tense vs past tense
- description vs action vs dialog
- standard literary language vs personal voice
There is no clear correlation between one extreme on any one of these dimensions and a reader's closeness to a character. Rather, they all combine to create an overall effect.
That said, let's consider your question.
A third person narrator is neutral. A third person narrator may stick to the perspective of one character or switch between those of several characters or even leave all characters and describe places or events without people. For example, in a story, the narrator may describe what goes on while all the characters are asleep:
The sun came up. The birds started to sing. Finally John awoke. Here, the first two sentences are "free of character", impersonal. There is no one there.
A first person narrator, on the other hand, clearly posits a character: there is "someone" there, and as readers we naturally focus as much attention on that character as the first person pronouns suggest that we do.
And there you have your answer. A first person narrator can bring themselves to the attention of the reader or avoid that attention, simply by how much they talk about themselves.
Look at this example, where I narrate a visit to the dentist with my son:
Yesterday, I went to the dentist with my son. On our way to the dentist, I could see that my son was apprehensive. As we got there, the dentist shook my hand, then greeted my son. I was worried, but I saw how my son relaxed, when the friendly dentist smiled at him.
Here, you, as a reader, will be confused about who the story is about: me or my son. Now compare this with another first person narration of the same events:
Yesterday, I went to the dentist with my son. My son was aprehensive and quiet on the bus, but when he saw the relaxed manner of the dentist and the dentist shook his hand with a big smile, he was no longer afraid.
Again, "I" am there, and you don't forget that I am, but I quickly step back and allow my son center stage. I can take this to an extreme, like Defoe in Robinson Crusoe, and appear only in the framing narrative. Or I can be a companion or side kick or the antagonist or another character with a more or less important role and continually take part in the story, and still not push myself before the third person protagonist.
"I", the first person narrator, can even do all that the third person narrator can, and narrate events that I did not witness. A first person omniscient narrator can be motivated (e.g. the protagonist later tells him what happened while he was elsewhere), or remain unexplained. Again, I want to remind you that the narrator is not a person but a device, and that readers will go with a first person omniscient narrator if that narrator is consistent and makes sense.