Learn English – What term is used for the closing of a letter

I have always heard the "Sincerely, Joe Smith" part of a letter called the closing.

(And the part you describe as the greeting ("Dear Mr. Blahblah") I'm used to calling the salutation.)

I personally think that it is a matter of choice whether you would like to use it or not. I'd say "Much appreciated" is a casual sign-off that should be used for expressing gratitude for favors, like trying to confirm with someone to be a reference for your letter of recommendation or basically asking of someone to do a favor, like babysitting, housekeeping, etc. and once the favor is accepted the response would sign-off with a "thank you" to redeem any sense of expressed gratitude as genuine. It wouldn't be taken as condescending because you have not received any response that would anticipate a declined offer. Essentially if you are starting a conversation asking for a request, "Much appreciated" is the best sign-off. When in doubt, it's always best to use "thank you" in place of "much appreciated".