Does it matter how you write it down? You can do it any way you want. There isn’t much of a difference between your approach and my approach.
Wikipedians do it the same way that I do except they start with (0).
And the reason we do it this way is because it’s more elegant.
Consider this (your approach):
(d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^0)
And now consider this (Wikipedia approach):
(d_n \dotso d_2d_1d_0 = d_n \times 10^n + \cdots + d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0)
Which of the two expressions look more elegent to you?
And this isn’t the only reason I write it that way. There’s another one.
As for why I start with (1) rather than (0), there’s a reason for that too.
Right. If (\infty) stands for “any number greater than every integer” then a number cannot be both greater than every integer and at the same time less than any number greater than every integer. That would be an obvious contradiction.
That’s why, as you already know, we have to do one of the following things:

we have to say that, in our case, neither (\infty) nor (\dotso) indicate “any number greater than every integer”, but rather, that both indicate one and the same specific number greater than every integer

we have to come up with a neutral symbol, such as for example (x), which we can say represents some specific number greater than every integer; and we also have to say that the number of digits in (999\dotso) is (x) (to make it clear, we can use something like (999\dotso_x))
Let’s pick the second way because it’s less likely to lead to confusion.
My claim is that (999\dotso_x) stands for (9 \times 10^{x1} + 9 \times 10^{x2} + 9 \times 10^{x3} + \cdots) where (x) stands for some specific number greater than every integer.
I also claim that 1) YOUR expression and MY expression represent two different numbers, 2) my expression represents a larger number, 3) my expression represents (999\dotso_x), and 4) your expression represents (\dotso999_x).
Now, let me take the following statement of mine . . .
. . . and reword it, like so:
The nonzero digits in (999\dotso_x) are digits whose index is less than (x) but greater than every integer.
But I think I made a mistake here.
Let’s see.
The index of the first digit is (1), so the index of the last digit is equal to the number of digits. Since the number of digits is (x), the index of the last digit is (x). Thus, the nonzero digits in (999\dotso_x) are digits whose index is less than (x + 1) but greater than every integer.
Do you agree so far?