You are writing a letter that purports to be written by a lawyer; you are pretending to be a lawyer that happens to have a different name than your actual name. That is both pretending to be a lawyer, and pretending other false things.
Consider if you change the communication medium from written to oral: if walk up to someone and claim a different name than is actually yours, claim to be a lawyer representing someone else (who you identify with your actual name), and then issue some demands, etc., it's pretty clear that you are claiming to be a lawyer. Changing the medium from oral communication to written doesn't change the essence at all.
I still don't see the argument, because I don't see where writing the letter translates to me claiming anything. Like, imagine that I wrote the letter as a text that would appear in a novel. And then someone else, reading my book, copy-and-pasted that text into an email and sent it to someone. I'm quite certain I'm not claiming to be a lawyer at that point (although the person who copied the text might now be, I guess.) What is the smallest change you can make to what I did by writing the novel, that would push it over the line into being a criminal act? When does a text change from fiction into a claim about reality?
I don't think the "walking up to someone" analogy applies, because there's a "me" doing that walking that the person can see, and so by asserting any claim through speech, I'd be attaching the subject of that claim to the person they're seeing—i.e. to myself. But in sending a letter written "as" someone else, there is no "me" from their perspective to attach the claim to. There's only "whoever is named [fake lawyer name]" (nobody) and "whoever sent this letter" (someone else, although I paid them to do it. Coincidentally, if I did have a lawyer, I'd be paying them to do it.)
> I still don't see the argument, because I don't see where writing the letter translates to me claiming anything
The letter is a written statement which (whatever else.you may claim in it) is, un fact, from you, asserting various facts, including that the writer of letter is a lawyer.
The claims in the letter are made by the actual author of the letter, not the (potentially fictions) person that they are claiming to be when they wrote it.
> When does a text change from fiction into a claim about reality
When it is presented either with intent that it will be taken as a claim about reality, or when a reasonable person would see it that way. What you write in a novel is an overt work of fiction that no reasonable person, in context, will, in the general case (barring special circumstances), see as a claim of fact. If someone else copies part of it out of context and presents it differently, thats a different story, but has nothing to do with you.
When you send something that you have written to someone with the intent that they view it as what it purports to be on its face, than any knowing falsehoods in it are knowing false claims made by you, the actual person who composed and sent the communication with the intent to deceive.
The key concepts that generally turn actions from innocuous to culpable here are: intention, and benefit. In particular - malicious intention or ill gotten benefit turn acts into the sort we as a society frown upon and hence attempt to deter with regulatory or criminal sanction.
Consider if you change the communication medium from written to oral: if walk up to someone and claim a different name than is actually yours, claim to be a lawyer representing someone else (who you identify with your actual name), and then issue some demands, etc., it's pretty clear that you are claiming to be a lawyer. Changing the medium from oral communication to written doesn't change the essence at all.