Jon, I hope I've understood your question, and that I'm not just rambling :) I don't know the name you were looking for, but perhaps I can say a few words on why the more flourished sign-offs have all but disappeared.
It started to drop away in 1950 (plus or minus a few years).
Many things changed after WWII. There was a major cultural shift away from the strict disciplinarian ways of the authoritarian structures which were seen to have precipitated the terrible world-wars. Egalitarianism was the catch-cry. Telephones were becoming a matter of course, and they were cheap. Television broke parochial boundaries.
Formality started to make way for familiarity.
Prior to 1950, writing a letter was a special moment in which you could reach out to otherwise unreachable friends and relatives.
Every word, every letter, was crafted as well and as thoughtfully as possible. The art of prose existed then; poetry was well appreciated then; oratory existed then.
Even the act of having to handle ink (by dipping or filling a cartridge), and the feeling of the bend and flex of the pen's nib was somehow personal and intimate.
A person's handwriting was a thing of pride and presentation. It was as unique as the person who wrote it.
Today it is all Times New Roman 12pt Regular, on a QWERTY keyboad... all rather mundane and boring and non-descript... and of course you can Tweet or SMS if you prefer.. or IM in "abridged English"... or bypass it all and voice-chat to anyone, anywhere, anytime...
PS: The wind-out you wrote above would more typically be as follows:
Thanking you in advance.
...A FLOURISHED SIGNATURE