Yes, you can use astral projection from planes other than the Material Plane. The Manual of the Planes describes how spells that access transitive planes such as the Astral Plane function on other planes in the section "Moving Among Transitive Planes" (starting on page 45).
Characters generally must use spells or spell-like abilities to access a Transitive
Plane. ... the astral projection spell takes you to the Astral Plane ... Such spells should function in any location coexistent with or coterminous to the plane.
It includes various examples of travel between transitive planes on page 46. These examples reinforce that "Material Plane" is a placeholder and that you can use astral projection from anywhere that is connected to the astral plane. Your body will be left behind on the plane you project from.
From the Ethereal Plane to the Astral Plane: You can move to the Astral Plane from almost everywhere on the Ethereal Plane, because the Astral Plane connects to everywhere on the Ethereal Plane. ... If you're using an astral projection spell, your physical body remains in the Ethereal Plane ...
From the Plane of Shadow to the Astral Plane: From the Plane of Shadow, you can use the astral projection spell, leaving your body behind on that plane ...
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.
Best Answer
D&D 5e made significant and substantial changes to nearly every major spell in order to accomplish its systematic rebalancing of the magic system. In all previous editions of D&D (except 4th, which completely replaced the entire subsystem), Astral Projection allowed you to visit other planes in such a manner that your death on the visited plane would leave no serious lasting harm. D&D 5e did, in fact, change that (at least in the spell description), but the author of the section on planar travel in the DMG is clearly writing with an understanding of the previous way the spell worked. It's quite probable that the person writing this section was a different author, and possible they had no idea the spell had been so significantly altered. In any case, I think the update in the PHB is consistent with the updates to magic overall, and should be preferred over the text in the DMG, which makes incorrect reference to the spell as a related topic.