I will answer for the "physical" aspect (other answers have addressed the "existence" aspect but it seems to me that what you're after is a distinction between physical and unphysical existence, not a definition of existence itself? Even if not my answer will complement the others).
The way you formulate your question makes me think that what you call "physical existence" is what philosophers call "concrete existence" (as opposed to "abstract"). Laws of nature (or numbers) are abstract while objects such as electrons and rocks are concrete.
Our most fundamental concepts are often difficult to define, but there are some candidate criteria for concreteness (all of which can be discussed):
- existing in space and time
- entering causal relations
- being mind-independent
- being particular rather than universal (as "red" vs "redness")
The first two are the most often used in contemporary literature. The third is controversial because some would say that abstract objects, such as numbers, are mind-independent too.
According to these criteria, the electromagnetic field is concrete: it exists in space and time, enter causal relations, is construed as mind-independent and it is a particular.
If now we turn to what philosophers understand by "physical", there are two main approaches:
- calling physical what is studied by physics (or an ideal, complete physics), and everything that reduces to it
- having distinctive criteria for physical things, such as: objectivity (mind-independence), reducibility/compositionality...
In any case physical objects will be a subset of concrete objects (or exactly the same set for physicalists), and again, I think we can say that the electromagnetic field is physical according to both approaches: it is an object of physics, and it is supposed to be objective and compositional (you can consider a part of the field in a specific region).
So in all cases it is safe to say that the electromagntic field physically, or concretely exists.
Let me end with a remark: it's not clear, having said that, that electrons or the electromagnetic field (or numbers or laws of nature) really exist. The doctrine that says so is called scientific realism (or platonism for abstract objects), but there are also anti-realist doctrines, such as instrumentalism or empiricism.
It is thus more appropriate to say that if the electromagnetic field exists, then it exists concretely/physically.
But as far as I understood your question wasn't about existence specifically but about distinct modes of existence, so that doesn't change my answer.