They are slightly different and they bind to different concepts of REST API (which is based on HTTP)
Just imagine that you have some Booking entity. And yo perform the following actions with resources:
POST - creates a new resource. And it is not idempotent - if you sent the same request twice -> two bookings will be stored. The third time - will create the third one. You are updating your DB with every request.
PUT - updates the full representation of a resource. It means - it replaces the booking full object with a new one. And it is idempotent - you could send a request ten times result will be the same (if a resource wasn't changed between your calls)
PATCH - updates some part of the resource. For example, your booking entity has a date property -> you update only this property. For example, replace the existing date with new date which is sent at the request.
However, for either of the above - who is deciding whether it is going to be a new resource creation or updating/modifying an existing one, it's the database operation or something equivalent to that which takes care of persistence
You are mixing different things.
The persistence layer and UI layer are two different things.
The general pattern used at Java - Model View Controller.
REST API has its own concept. And the DB layer has its own purpose. Keep in mind that separating the work of your application into layers is exactly high cohesion - when code is narrow-focused and does one thing and does it well.
Mainly at the answer, I posted some concepts for REST.
The main decision about what the application should do - create the new entity or update is a developer. And this kind of decision is usually done through the service layer. There are many additional factors that could be done, like transactions support, performing filtering of the data from DB, pagination, etc.
Also, it depends on how the DB layer is implemented. If JPA with HIbernate is used or with JDBC template, custom queries execution...