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Is EdgeDB going to be Postgres fork, extension or a service that uses just uses Postgres as a storage layer (like Datomic)?

It's closer to the storage layer: postgres is in the core and not directly exposed (i.e. you won't be able to access it).

Does it mean it will be possible to use EdgeDB with hosted Postgres databases, like RDS and maybe Amazon Aurora (on Postgres mode)?

Maybe. This is something that we'll have to figure out later.