"Embedded DSL" is for better or worse a reasonably well established term for DSLs that are embedded within a host language using the language's own abstraction constructs (libraries, macros, etc.), rather than being implemented as a separate language with a parser and compiler/intepreter. I do usually use the alternative term "internal DSL" though, which I believe is also the older one. Besides avoiding the name clash with embedded programming in the low-level hardware sense, I like it better because the opposite of an internal DSL is an external DSL, not an "unembedded DSL".
An EDSL and a domain-specific API are not the same thing. For example, one typically expects the description of a program's logic and its execution to be separate in a DSL (i.e. building up expressions from the syntax made available in the EDSL and then later applying an interpreter (possibly multiple interpreters) to these expressions) while this is not generally true for domain-specific APIs.
I do in fact have a horse in this race as an Embedded developer, but I don't really ascribe worthiness of exclusive use of the term to any particular side. Theyre all just using a term that predates all of their industries, so this isn't a thing to pick or defend sides on, in my opinion. I simply lament the inability to be specific and comprehensive. It's not like "Embedded" is even a hot buzzword or has anything resembling unique brand recognition.
