If a definition is not meant to be explanatory, its usefulness in understanding that which is to be defined is limited.

Taking the two alternate formulations you offered, I can still hear the telltale "is" beating, from beneath the floor planks where you hid it:

'exists(a)' ≝ '¬∀x¬(a=x)'

The "∀" doesn't refer to all e.g. logically constructible x, does it? Or to all computable x. For the definition to make sense, it needs to refer to all x that exist, otherwise we'd conclude that 'exists(flying unicorns)' is true. Still implicitly refers to that which is to be defined in its definition, rendering it circular.

'exists(a)' ≝ 'a∈EXT(=)'

What is EXT(=)? Some set of all existing things? If so, would that definition do any work for us? Pointing at my chair and asking "does this chair exist", you'd say "well, if it's a member of the set of all existing things, it exists". Why, because all things in the set share the "exist" predicate. But what does it mean for them to have the "exist" predicate in the first place? To be part of the set of all existing things, of course. Round and round ...

Not much different from saying "if it exists, it exists". Well, yes. Now what?