Ah, so they do. I was in error.
Is fiend a superset, which includes the set of Infernal creatures and the set of Abyssal creatures?
If so, then are there fiends other than Abyssal fiends and Infernal fiends?
Do any of the 5E books define any of these terms? DMG describes the Abyss, in the section on planes.
Fiend is a superset describing the evil supernatural beings native to the Lower Planes (Acheron, Nine Hells, Gehenna, Hades, Carceri, the Abyss, and Pandemonium). The term is briefly described in the MM along with the other creature types (and I mean very briefly).
Infernal is a term generally associated with the lawful evil fiends (devils--which are generally based out of the Nine Hells of Baator), while Abyssal refers to the chaotic evil fiends (demons--hailing from the Abyss). Those terms also refer to the respective languages of the devils and demons.
There are plenty of other fiends, although only a few are represented in 5e. The Yugoloths are the major neutral evil players in the fiend field and originally came from Hades, and then many of them migrated to Gehenna (although the 5e MM made some intolerable changes to the lore--in my game I'll likely say that the MM story is the version of the story that the devils tell). In 1e, yugoloths were known as "daemons."
The MM addresses a scattering of other fiends. You have the LE rakshasa, and some miscelaneous critters like hell hounds and nightmares, as well as night hags. Night hags are also big players in Hades.
There are/were (depending on whether you wait for WotC to print it before it exists in your world) a variety of other fiends connected to various lower planes. Hordelings were these cool fiends for running into in mobs in Hades. The gehreleths (1e demodands) are the native fiends of Carceri--the prison plane. They are both jailors and prisoners. Those are the ones that are hitting me off the top of my head, but there were some other lesser fiends.
There is really a lot of cool stuff about the various inhabitants of the planes. Some people don't really care about the various races and their differences (how many types of "demon" do we need?) but if you are someone who appreciates that sort of thing (and I do), it can be very interesting to delve into what makes each race unique and how they relate to the rest of the multiverse. Gehreleths provide an entirely different vibe than baatezu (devils), for instance. While they look different, the gehreleths give me a bit of Hellraiser feel. This richness of inhabitants isn't limited to just the evil races either. You have a variety of different good races based on different planes and alignments also (none of which, other than a few angels, are yet published or acknowledged in 5e material).
If you are interested in knowing how it all got developed, I'd refer you to the free 3e Planescape Campaign Setting downloads over at
http://mimir.planewalker.com/sections/30-35-pscs.
And if you like that, you can always buy some of the older material at dndclassics.com. If you want a great overview (that's how I operate, but I know a lot of people don't, so it's preference), the 2e Planescape Campaign Setting, 1e Manual of the Planes, and 3e Manual of the Planes are good stuff.
So that's probably more than you wanted to know, but I hope it's helpful!
