Is this rewrite correct? If x mod 9 = 1, can I rewrite the equation as x = 1 mod 9. If not, what is the correct way?
I know this sounds silly. But please help me.

There are two possible meanings of "mod" here:
One is a binary operation: $a \operatorname{mod} n$ is a number; it is the unique integer $r$ with $0 \le r < n$ such that $a - r$ is a multiple of $n$. If interpreted this way, the two equations aren't equivalent (for example, $x = 10$ is a solution of the former but not he latter).
The other is an equivalence relation: $a \equiv b \pmod{n}$ is a statement; it means that $b - a$ is a multiple of $n$. Only the second equation you wrote makes sense in this notation.
Maybe you could mix notations (e.g. it is always true that $a \equiv a \operatorname{mod} n \pmod{n}$), but that rapidly gets confusing.
