Saying something is 9 times bigger is also a shorthand because “times” is a shorthand for repeated addition. You are adding a number “9 times” to make it “9 times larger”. But we don’t say the “adding” part.
Likewise if I repeatedly subtract a number 9 times to make it “9 times smaller” I can omit the subtracted bit.
That interpretation doesn’t make sense. Repeated subtraction does not correspond to division in the way repeated addition corresponds to multiplication.
This isn’t analogous to repeated addition giving multiplication. You’ve switched from applying an operation nine times to counting how many times an operation can be applied.
You chose 81 because it’s the one number that makes your argument appear to work. Try the same reasoning with 80 ÷ 9.
Likewise if I repeatedly subtract a number 9 times to make it “9 times smaller” I can omit the subtracted bit.