As someone from Kentucky, there’s a difference between The Beach and A Beach. Colloquially, the beach is with the ocean but a beach is anywhere with sand where you swim. Lots of our lakes have beaches. A beach doesn’t even have to have sand, basically just be formally cleared for the purpose of swimming/etc.
I’ve only ever heard “going to the shore” as someone coming in from the water, not going out to it. If you’re swimming in the middle of the lake, you’d announce that you’re going to the shore, but you’d never say that abt driving from the city to the lake. Same concept with The Beach afaik