On page 67 they are listed at #9 on the infernal hierarchy in the ranks of the greater devils. This is part of the general section on the Infernal Hierarchy.
Along with this text also on page 67.
Greater Devils. The greater devils include the pit fiends, erinyes, horned devils, and ice devils that command lesser devils and
attend the archdevils.
So regardless of their origins the designer's intent is that they are part of the Infernal Hierarchy.
And note the exact text on page 69 is
Legends tell that the first erinyes were angels that fell from the
Upper Planes because of temptation or misdeed
So at first the erinyes were fallen angels and probably the oldest and most powerful still are. But this implies are also erinyes that were promoted through the ranks.
In the case of demons, the monster description (MM p. 50) argues against the reverse of falling that the angels are susceptible to.
Possessing no compassion, empathy, or mercy, they exist only to destroy.
Angel: pride can lead to a mistake that leads to a fall. (MM p. 15)
Demon: opposite of angel ~ the opposite of pride is humility.
Humility doesn't fit the profile of lacking empathy, mercy, and compassion. From that starting point, the DM needs to establish a motive that outweighs this fundamental characteristic of a demon. Something, some experience, someone, or some event induces a demon to act in an other than selfish and destructive manner.
The DM then must resolve how a creature spawned as an extension of the Abyss (MM p. 50, Spawn of Chaos) breaks free of that influence sufficiently to change its nature. In the case of demons formed from mortal souls (since in 5e souls exist) the memory of being something other than a demon could provide the leverage needed to initiate change. In that case, the specific follow-on effects will vary as with the motive for change.
Two simple and opposed potential outcomes:
- The demon leaves the Abyss forever, never to return
The demon enlists allies to return to the abyss and avenge ---
something.
(And many points between). Without a story hook on the why of a demon's change, the what of the aftermath of the change is too varied to nail down.
MM p. 7 (RAW boils down to "You can if you want to.")
"Feel free to depart from it and change a monster's alignment to suit
the needs of your campaign. If you want a good-aligned green dragon,
or an evil storm giant, there's nothing stopping you.
The only thing stopping the DM is the crafting of an in-world reason that fits the larger narrative of the campaign and the story. The only constraint is the imagination of the DM, not a rule.
What makes for a good change is a decision to create the exception to the general rule of the MM default description for demons - the general case is that a demon isn't motivated to change. Establishing motivation is step zero to this change occurring in, or fitting the narrative of, a campaign.
If lore extends beyond 5e, something from 2e Planescape or FR novels may offer a concrete example.
There were a few risen fiends in 2e Planescape - Fall-From-Grace the succubus, and Morte the floating skull whose provenance is something of a spoiler ~ {Thank you @nick012000}
Best Answer
A devil is formed/born when (MM 66, top paragraph)
So, answering about Rakshasas and Hell Hounds, they are each a unique fiend rather than being a form of corrupted mortal soul.
On the next page is a list of four main categories of devils, with a subsection called Promotion and Demotion covering how a corrupted soul can get from newly minted minion all the way up to the top levels of the Infernal Hierarchy.
In the stat blocks on MM 70-78, it appears that all Devils have the following features:
Most also have natural armor and Devil's Sight (for some reason, not Pit Fiends or Erinyes).
Making Your Own Devil
If you're making your own devil then the only real core is Lawful Evil and some kind of corruption of a mortal soul. I see two options to make it interesting.
First: Backstory.
A favored NPC is a literal devil in disguise. The drama arises from the emotional connection PCs have to this NPC. This is basically a literary classic.
In D&D terms, though, you might consider what happens differently when an Elf or Dwarf is corrupted, as opposed to a Human. The books don't seem to say anything on this, but it's totally within DM purview to build the world in this regard.
Second: Custom monster.
DMG 273 details how to create or modify a monster.
My favourite technique for this is covered in the Modifying a Monster section. Just take a different monster and describe it differently. A short step away from that is describing a creature differently and also tweaking small things like energy type on damage. For example, you could just take a Deva, give him red skin and horns, then tweak that Radiant damage to Necrotic. There's a section on Fallen Angels on MM P. 15 if you'd like to commit to this route.
If that's not enough, then use the subsequent sections to create your own. The most important step here is in the opening paragraphs: the concept. If you're sticking with the book's 'corrupted soul' concept, then consider what sins/corruptions might have created the various devils in the book. Then consider what your new creation's sin might have been (backstory again!) and what thematic powers, traits, and quirks might manifest. Add those to the above list of generic devil traits. Additionally, you should consider how this devil fits into the Infernal Hierarchy, including which devils owe it fealty and which it owes fealty to.