Salem

Well, the question is in the title, can you get lobsters from rivers that start from the ocean or just by the beach?

I don't think rivers start from the ocean
Actually they flow to it, so get your conclusions

I think that even if you fish/trap at the ocean shore but there's a river exiting there as well it will count as a river not an ocean, so no, the water biomes are very simple so don't overthink it

You can catch ocean fish from the river going through providence though. I've got chests of cod fished out of the prov river.

easy to test. you can't fill buckets from rivers inside prov. you have to go a bit back, like the small lake at the west end of prov.
so, where you can't fill a bucket you should get lobsters