So after comparing the definitions, I have no idea why these seem to be defined so differently.
A space $X$ at $x$ is locally connected if there is an open set $U$ containing $x$ such that there we can find a open connected set $x \in V \subset U$. I guess in other words, the open connected sets $V_x$ form a local base.
A space $X$ at $x$ is locally compact if there is a compact neighbourhood $C$ at $x$.
It looks like to me that local compactness is roughly saying a space $X$ is locally compact on some point $x$, if on some neighbourhood $U(x)$, $X$ is compact on that neighbourhood. In other words, $U(x) \cap X = U(x)$ is compact, which seems to agree with what one might normally come up with for the definition.
On the other hand, local connectedness says we can find a connected neighbourhood inside an open set...
I've read some answers from MO, but none of these feels satisfying to me (a lot of red-herring answers I feel...). I just don't understand what's wrong with defining local connectedness as having "a connected neighbourhood" or local compactness as "a compact set inside an open set". Maybe there are spaces that simply do not have connected (or compact) subsets?