Are there any concrete examples of $\sigma$-algebra generated by a random variable?

I have searched around the internet for any concrete example of $$\sigma$$ algebra generated by a random variable $$X$$ but failed to find any nontrivial, concrete examples.

For example,

https://stats.stackexchange.com/questions/312474/what-is-it-meant-with-the-sigma-algebra-generated-by-a-random-variable

Consider a single tosses of a fair coin,

Then the sample space is $$\Omega = \{\{H\}, \{T\}\}$$, with $$\sigma$$-algebra on $$\Omega$$ equal to$$\mathcal{F} = \{\{\varnothing\}, \{H\}, \{T\}, \{H,T\}\}$$

We can define a random variable $$X$$, where $$X(H) = 1$$, $$X(T) = -1$$

Now I ask, what is the $$\sigma$$-algebra generated by $$X$$?

By definition,

$$\sigma$$-algebra generated by $$X$$, denoted $$\sigma(X)$$ is the collection of sets $$\sigma(X) = \{\{\omega \in \Omega, X(\omega) \in B\}: B \in \mathcal{B}\}$$, where $$\mathcal{B}$$ is the Borel set.

(Is my definition correct?)

Let's pick some sets,

$$B = \{1\}, \omega = \{H\} \in \Omega$$

$$B = \{-1\}, \omega = \{T\} \in \Omega$$

For $$B$$ excluding both of these singletons, $$\omega = \varnothing \in \Omega$$, and for $$B$$ including both of these singletons, $$\omega = \Omega \in \Omega$$

So the $$\sigma$$-algebra generated by $$X$$ is the same as generated by the subsets of $$\Omega$$.

Am I right in my reasoning?

What is a concrete example where $$\sigma$$-algebra generated by $$X$$ is not the same as that generated by the subsets of $$\Omega$$?

For example, you might try $$\Omega = \{-2, -1, 0, 1, 2\}$$ and $$X(\omega) = \omega^2$$. Note that $$X$$ takes the same value on $$\omega$$ and $$-\omega$$, so any member of the $$\sigma$$-algebra it generates contains either both $$\omega$$ and $$-\omega$$ or neither of them.
For any $$A \subset \Omega$$, $$\{\emptyset, A,A^{c}, \Omega\}$$ is a sigma algebra and it is generated by $$X=I_A$$.