all the Alter characters are, as you say, alternate versions of their originals with some having alternate legends.
however the idea of Alter extends from a canon character which is Saber-Alter from Fate/Stay Night who makes her appearance in the Heave's Feel Route after
Saber is consumed The Shadow. by returning to the Greater Grail she is corrupted by Angra Mainyu and first appears when Shirou, Rin and Ilya are fleeing Blackened Berserker and operates as an alternate Servant for Sakura
Outside Fate/Stay Night Alter Saber acts as an "alternate form" of Saber which comes out when Saber's ahoge is plucked in Fate/Hollow Ataraxia
despite being an antagonist in Fate/Stay Night Gen Urobuchi saw Alter Saber as a Dark Hero
Uro: What I imagine is that for the Saber Lily storyline, in Fate/stay night, Saber was completely beaten by Caster, and so she became one of her Servants. Like, she thought "I'd be way better off with Caster than with Shirou." (Laughs)
Nasu: I never thought of that. Interesting. So in other words, the white dress represents how both her body and soul have been tamed by Caster. So Lily is truly like a "lily" to Caster... you crazy bastard, you did it again...
Takeuchi: Lily's design was inspired by the dress she wore when Saber was captured by Caster. Then, we just filled it with feminine qualities that Saber lacked and it became what it is now. So basically, I think Urobuchi really hit it close to the mark.
Uro: Really?! Lily looked like a villain to me from the moment I saw her. Imagine her appearing from behind Caster, really working those heels! Just thinking of it gets me excited!
Takeuchi: But in terms of villains, don't we already have Alter?
Uro: No, she isn't a villain, she's a dark hero.
Nasu: Alter is an idealist who clings to her own values even though she's been blackened. In that respect, she isn't a pure villain.
Uro: For example, Alter's like how Hakaider was in Mechanical Violator Hakaider (*8). And Lily is the enemy she would face. The wrinkles in the corners of her eyes are proof to that. She'd shine as a villain.
Nasu: Then after defeating Lily in battle, Alter would jump on a motorbike in the last scene and ride off into the sunset. Then the narration would come in: "She is a modern knight, who lives in a world of darkness!" (*9)
Takeuchi: Motorbikes aside, (laughs) I do think it'd be interesting to add meat to Lily's character in that manner.
Source: Saber Alter - References 5 - TYPE-MOON Ace 7 - 10 year Anniversary interview: Nasu Kinoko X Takeuchi Takashi X Urobuchi Gen
before Fate/Grand Order other characters got Alters were sort of jokes created by Takashi Takeuchi and sometimes weren't even separate people such as Sakura Saber who's "Alter" was her Nobel Phantasm Coat of Oaths but after Fate/Grand Order the started becoming real
Alter is actually one of 3 variations of a hero themself
- Alter (オルタ, Oruta?) - Servants greatly altered from their normal selves through various means are often given the title of "Alter" after their name. True Archer, an altered Heracles, rejects being called "Heracles Alter" and instead chooses his "human name" of "Alcides".
- Lily (リリィ, Riryi?) - Servants who take on a more youthful appearance compared to their older selves are often given the title of "Lily" after their name. This title is normally only applied to female Servants, but those like Child-Gil and Alexander are referred to as Lily Servants elsewhere.
- Santa (サンタ, Santa?) - Servants acting as Santa in the Fate/Grand Order Christmas events are given the title of "Santa" after their name.
Source: Servants - Alter
and as such shouldn't be confused with Alter Egos which are a separate Servant Class where Alters can be any class
The current list of Alters (as of 19/01/2018) aside from Alter Saber from Fate/Stay Night are
- Alter EMIYA - a disgraced heartless version of the Counter Guardian EMIYA who's cast off his name
- Alter Lancer Arturia - an Alter form of Lancer Arturia who, before changing into the "Goddess of the Holy Lance" chose to insated become the tyrant known as the "King of Storms"
- Beserker Cu Chulainn - a version of Cu Chulainn reflecting the wish made upon the Holy Grail by Queen Medb
- Mysterious Heroine X Alter - a villain from the Servant Universe who has sworn to defeat Mysterious Heroine X
- Avenger Jeanne d'Arc - a fabrication of Jeanne d'Arc created by Gilles de Rais via the Holy Grail after her death
- Jeanne d'Arc Alter Santa Lily - Avenger Jeanne d'Arc after drinking Gilgamesh's Potion of youth (becoming "Joan of Arc Alter Lily"). gained the title of "Santa" when she was aiming to become "Santa Joan of Arc."
- True Archer - the human side of Hercules who lost his divinity after Bazdilot Cordelion used his Command Spells to awaken memories from the 5th Holy Grail War and his corruption in it. he refuses to be called an Alter Servant and takes the name Alcides instead
and as explained above, Sakura Saber's Alter appearance is actually her Nobel Phantasm, Coat of Oaths