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 ...
The halberd and glaive are there because D&D has a history of offering a wide variety of codified polearms. Namely 2e and previous, which 5e strives to emulate in many regards.
It's fairly likely that someone on the design team, or if not them, someone that someone on the design team talked to, thinks that D&D is not D&D without some variety of polearms to choose from. This could be just "one of those things" (commonly called "sacred cows" in jargon) that becomes a tradition of the franchise and outlives its usefulness by virtue of the fact that many players are familiar with it.
To demonstrate my point, it's easiest to again refer to the 2e Arms and Equipment book, as @nitsua60 did, but also in addition to the core rules. Here's a list of the different polearms codified in the core rules and A&E:
- Glaive
- Halberd
- Lucerne hammer -- like a halberd, but with a hammer.
- Guisarme -- a peasant's weapon specialized in dismounting knights. It's defined by a hook on one side, usually with a spear tip emerging from the hook as well.
- Longspear
- Ranseur -- a spear with a crossguard, like a trident but with unsharpened, shorter side-points.
- Scythe
- Trident
This is pretty exhaustive even considering D&D, and even considering that it's spread over two books. Remember, we're dealing with a subset of two-handed melee weapons. 3rd edition, as most are well aware, did not make any meaningful attempt to curb the amount of codified rules. Pathfinder continues the tradition; a quick glance at the Weapons table in the d20pfsrd confirms that, even going so far as to make the arguably-pointless-in-real-life distinction between the bec de corbin and the lucerne hammer.
In fact, D&D's codification of long weapons goes all the way back to OD&D 1e at latest. Gary Gygax included Appendix T to AD&D 1e Unearthed Arcana (1985) which was an extensive discussion of pole arms that included citation to four different text books about medieval weapons. He ended up including thirteen (that is 13) different varieties of polearms in the original D&D 1st edition Player's Handbook. Even before that, Gygax had provided polearm supplement rules for Chainmail, D&D's wargame predecessor, via the wargame magazine Strategic Review, second issue. Many thanks go to Korvin Starmast for providing the information.
Things like this seem simple and pointless when viewed from an outsider's perspective, but it's just one of those things that gives D&D its character, something that sets it apart from other games, even if it is only a small thing. Many people are sad to see those defining characteristics go.
Best Answer
In 5th Edition, the primary source of information about both planes is the DMG. Since you're asking about 5th specifically, we'll avoid talking about earlier editions.
The Ethereal Plane is what connects the Prime Material Plane, where all elements are found, to the Elemental Planes, which serve as the sources of those elements. While in the Ethereal, you can perceive the Material Plane if you're close enough to it, a region known as the Border Ethereal. Think of the various planes of matter and energy as a sphere, with the Prime Material as the globes surface and the different Elemental Planes bobbing about inside it. A mirror image of the Prime Material serves as the Feywild, and the shadow the Prime Material Plane makes is the Plane of Shadow.
The Astral Plane, on the other hand, is a timeless void. Everything in existence (except, due to the weirdness of cosmology, the Inner Planes) is accessible from the Astral Plane. Unlike the Ethereal Plane, travel in the Astral Plane is tied directly to thought, and it is believed that intelligent creatures travel to portions of the Astral Plane as they sleep. Near the outer areas of the Astral Plane, itself an eyeboggling concept due to its infinite size, are the realms of the abstract where the gods live and souls travel upon death, the Outer Planes.
Your key takeaways:
The Ethereal Plane can look upon the closest plane. It is otherwise a foggy realm that connects the Inner Planes and the Material Plane. It's not true to think of the Inner Planes as inside the Material Plane and the Ethereal as the goo holding it all together, but it helps.
The Astral Plane is timeless, almost featureless, and is home to everything non-physical or beyond mortality. The Astral Plane is what the Ethereal Plane sits in, surrounded at a great distance by the Outer Planes.