The beachhead concept, as outlined by Geoffrey Moore*, is about finding your first solid foothold in the market. It's also often used to suggest your initial target segment.
Even within that segment, the question arises, how should you design an MVP to make the most impact and for who? This is where the MVS comes in.
The MVS, delves deeper. It's about finding a group of customers with a shared need or set of needs.
It might be a sub segment of the beach head, but in any case, the goal is to laser focuses the tiny resources of a startup in a way they can win over customers repeatablywithoutchanging the MVP, because the customer needs are the same.
Whereas in a segment, or beachhead, unless it is more tightly defined there is no guarantee potential customers will have the same needs just because they are for example all small and medium sized businesses in the same industry. It may depend on how they operate, who their customers are, how they serve them, what their business model is and so on. For more see here
*Note I was lucky enough to have Geoffrey on my advisory board and many years later invited him to the Harvard i Lab for one of my Startup Secrets classes. I have the utmost respect for his work and indeed the prior work he built on