The expected pattern for a formal letter to close the correspondence with the name of the correspondent.
But, sometimes people write their names in the body of the letter as well.
Is it superfluous to write your name and other information in the body of the letter, as we must write it at the bottom anyway? As in should the body of the letter only use appropriate pronouns (I, we, us, and 'the party of the first part') when referring to the correspondent?
A formal business letter should always be signed at the bottom by the writer, and the name (and possibly title) should be typed or printed below the signature. This is true even if the name is stated in the body of the letter. The address, but not the name, of the sender are usually given in the heading.
A personal but formal letter should be signed with one's full name, but printing the name below the signature is less common. However it should be done if there is any reasonable chance that the recipient will be confused on who is sending the letter.
If the sender is unknown to the recipient, and particularly when the sender wants an organization to check the proper file it is often wise to start with a name. For example
Commissioner Jones:
My name is Jane Doe. My Contract number is 2124789. I want to bring to your attention ...
...
Jane Doe
It depends on how formal the letter is. It is often the case that the correspondent states their company name and their title at the top of the page as part of the heading. In this case, you are correct in supposing that pronouns would be used. For example:
Dear Mr/Ms ___,
We write to inform you that a building inspection will take place at your address on 5th April 2021. Our representative will call at approximately midday and will telephone you beforehand in order to provide you with the exact time of his call.
We hope this does not inconvenience you and please contact us if you have any difficulty with this arrangement.
Yours, ___
So, yes, in this case, pronouns throughout.
If the purpose of the letter is to introduce yourself (as a job applicant, as the new owner of something this person uses like a gym or an apartment building, as a local political candidate) then you should include your name in the body.
My name is Somebody Something and I believe I would be an excellent ...
If the letter serves any other purpose, and the recipient either knows who you are or doesn't care (eg notifying tenants of a fire drill) then I/we in the body is just fine, no matter how formal the letter.
I would not use "party of the first part" anywhere other than a contract or a letter that serves as an amendment to one (eg a letter confirming that the condition in a conditional offer has been met) -- these are generally heavily templated so I would just follow the template.
