I understand the definition of limit. I can use it to prove things, but I still don't get the motivation behind it.
As someone who took intro real analysis last quarter, I can relate.
"definition is somehow not capturing the intuitive idea of limit as a variable moving towards a value."
It does, I just think you need to think through the definition more, perhaps see some visuals. The key is word is $\forall \epsilon > 0$. Imagine starting from a large $\epsilon$ value. You can then find a $\delta$ in the domain for which the definition holds. Now imagine shrinking $\epsilon$ by a little, you can find another $\delta$ which works. Keep shrinking $\epsilon$ until it is super small. You can still find a $\delta$ and a corresponding region for which the definition holds.
This visual might help. https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit . Just imagine shrinking the $\epsilon$.