Showing that Lipschitz continuity is a subset of uniform continuity isn't hard.
Let's assume that it is true that $|f(x_1) - f(x_2)| \leq K(x_1-x_2)$ given an arbitrary value $\epsilon > 0$ let's define $\delta = \epsilon / K$
Then trivially if $|x_1 - x_2| \leq \delta$:
$|f(x_1) - f(x_2)| \leq K\cdot|x_1 - x_2| \leq K\cdot\delta \leq K\cdot\epsilon/K \leq \epsilon$.
Which satisfies the uniform continuity property that for every $\epsilon$ there is a $\delta$ such that $|x_1-x_2| \leq \delta \implies |f(x_1) - f(x_2)| \leq \epsilon$
However, I am having a hard time trying to understand why not all uniformly continuous functions obey this property.
I would appreciate a general explanation with at least one example of a non Lipschitz continous function that is uniformly continuous.
Although I admit I had not found the answer that is linked. That answer, although very good, still does not suffice. The author of the question provided multiple functions that are not Lipshitz continuous, but did not prove why they are not Lipshitz continuous (although he did prove that they are uniformly continuous).
My issue is not an absence of an example, but rather a lack of intuition to understand the difference between both.