what does $|x-2| < 1$ mean?

The geometric interpretation, in $\Bbb R$, for $|x-a|<b$ is '$x$ is at a distance smaller than $b$ from $a$'.

In your particular example, $|x-2|<1$, it means that $x$ is at a distance of at most $1$ from $2$ and it (the distance) never reaches $1$.

To interpret $|x-2|=|-x-2|$, I find useful to first note that $|-x-2|=|x-(-2)|$ (why?). The equality $|x-2|=|x-(-2)|$ says that $x$ is at equal distance between $2$ and $-2$.

More generally, $|x-a|=|x-b|$ says that $x$ is at the same distance between $a$ and $b$.

To summarize, read $|x-a|$ as the distance between $x$ and $a$.

Hint

Denote $x-2$ by $y$ then $$|x-2|<1\iff |y|<1$$ and you find exactly your first inequality. Can you take it from here?