Sadly the rules are, if not silent, very quiet on this issue. In 3.5e this was handled by the Extraplanar subtype that creatures acquired when they traveled between planes and the rule that any creature that was not Extraplanar defaulted to being a native of the material plane. 5e is a brave new world though so we can't just use those rules.

Native, or indigenous, species are species that exist in a given ecosystem without being introduced by human activity. This makes sense in our world where the term is generally used when distinguishing between parts of an ecosystem disrupted by human presence and parts that we haven't gotten to yet.

So how about D&D? In D&D the term, "native to this plane of existence", could mean, "is meant to be on this plane of existence and is only found elsewhere due to intelligent intervention". Let's look at how that works out.

• Celestials, Demons, Devils, all have their own planes to be on even though they often leave them.
• Elementals the MM specifically says they are native to the elemental planes and that matches our definition since they only leave if someone summons them.
• Aarakocra are interesting as despite being humanoid they are indigenous to the plane of air. They also exist on the material plane, but it is mentioned that they still hang out near portals to the plane of air.
• Slaads are an interesting case because while they live in Limbo they need humanoid hosts to reproduce and those are in short supply there. I'd say Slaad are actually native to both the material plane and limbo since they appear to commonly use both for their life cycle.
• A human shape-shifted into a fire elemental is still a human for our purposes and so native to the material plane. Sure they look like a fire elemental, have the powers of a fire elemental, and are near indistinguishable from a fire elemental, but it's only due to (hopefully) intelligent intervention that they got that way.

Using this definition leads to some tricky questions but generally works. Importantly it seems to match the intention of the spell to send creatures back where they belong.

No

To open, lets look at a sample of the rules that say you cannot permanently kill such a creature outside of their home plane...

The only way to truly destroy a demon is to seek it in the Abyss and kill it there.

MM 51

Devils that die in the Nine Hells are destroyed forever

MM 67

Only on its native plane can a yugoloth be destroyed permanently

MM 311

Of note: Angels in the 5E MM do NOT have this protection. Weird as it may seem, it is actually easier to kill a Celestial than it is to kill a Fiend.

The nature of Astral Projection is that your Physical Body and your Astral Body are separated. Your Physical Body is left behind on your 'Home' Plane while your spirit goes walkabout.

Bringing Fiends and Celestials to the Prime doesn't work like that. When you summon one, it isn't just their spirit you are summoning while their true body is left behind on their home plane, you actually summon them.

The descriptive text for Demons on MM 50 says

Wherever they wander across the Abyss, demons search for portals to other planes.

The descriptive text for Devils on MM 66 says

Devils are confined to the Lower Planes, but they can travel beyond those planes by way of portals or powerful summoning magic.

In both of these cases, it is explicitly mentioning 'portals' as a way that Fiends find their way to the Prime. Portals are a way to physically move from place to place, not a way to send your spirit somewhere without your body tagging along.

Gate, in essence, is a portal that you create between where you are and anywhere else that can suck something through if you know their True Name.

Thus, a Demon who found a Portal to the Prime Material Plane and a Demon who was called to you through a Gate both, effectively, arrived on the Prime in the same way. They are both entirely and physically there. And because they are not on their Home Plane...they cannot be permanently killed.

Simply put...Astral Projection is the exception to interplanar travel rules - it is not the general rule. Plane Shift, which physically moves you to another plane, is only 7th level--compared to the 9th level Astral Projection. Astral Projection is special because it gives mortals the same degree of insurance when plane-hopping that Fiends and some Celestials enjoy naturally.

Just as a cautionary note, though: if you are Astrally Projecting, watch out for 'Silver Swords' (most commonly wielded by the Githyanki). Those can still kill you for real.

To sum up:

tl;dr:

The books say that a Fiend cannot be permanently killed on a plane other than their home plane. How they left their home plane doesn't matter.

Astral Projection is an incredibly powerful spell because it lets an Adventurer mimic that same feature via magic.