Soul - the entirety of the information of a person recorded by the world system. Fixed/anchored onto a physical body and given a time limit of existence in the material plane according to that.

Based on what a person does, it's not just humanity that will remember it, but the world (the overall consciousness of the astral plane) itself will.

Heroic Spirits - souls/records that are removed from the cycle of transmigration and incorporated/anchored/frozen into a system that transcends time and space.
A soul that has reached the point of being remembered based on the magnitude of the alteration of destiny. (according to Rakkyo, while the earth is on the verge of destruction everyday, there are people preventing this all the time, which makes it "harder" to be considered a hero; in other words destiny either isn't being altered or is already too convoluted).

Servant class - a vessel and loose definitions for past heroes that makes it easier for an aspect of a soul (record) to appear/(be reproduced) as it had been back when it had been alive. Maybe a better way of putting it is as a simulation of a past record instead of an extension (which is what the Third Magic is supposed to do).

The purity of that soul/record is limited by what had parts of it had actually been recorded and emphasized. The materialized information of lesser Eirei won't be recorded 100% faithfully because the only information in the Throne left are only the parts that have been recorded/"remembered".

The base personality and memories exist but the fact that people remember it, rather than the fictional embellishments added on to it, ensures that the recording of the "information" that is translated into a materialized spirit body is conveyed with accuracy.

The "information"/legends that aren't experienced by the original hero during life exist only as vague information/knowledge instead of additional information/powers carried on to the soul.

A Hassan will have no identity other than the role he played as a Hassan. Exaggerated physical traits (like size) may or may not occur (aka the dwarfish Hassan; but Bazett thought it was just as possible that he could have been like that IRL).

Medea, while obviously capable of using healing sorcery for non-selfish means, won't be able to use it successfully for purely selfless intentions because people never remembered as using her powers in that way.

You could have the same Heroic Spirit divided into 2 Sabers for instance, which would emphasize two extremes. For instance, the record of one Saber being filtered towards the more negative aspects and the record of the other being filtered towards the more positive ones.

And the real Avenger, not the hollow ataraxia Avenger, was supposed to be no stronger than a normal human being, despite the fact that people "remembered" him as a god/the king of demons. The only other record/information that had been faithfully kept was the fact that his very existence had been made "evil". Which is why the normal Avenger was just "nothingness" given a humanoid outline with a soul that had a natural inclination towards evil (because the only thing that was left of him when he had been alive was a soul that had nothing left but hate). If Servants actually gained powers (other than class abilities/stat modification by regional worship/legends and a spirit body that can't be affected by normal weapons), you'd think that Angra Mainyu would be just as powerful or at least have maxed stats within the limitations of the Servant system.

Mana/maryoku involvement in Servants - how faithfully the materalized Heroic Spirit can make use of the powers he/she had back when he/she was alive.
There's a note in the Black Saber description implying that it's not so much the legendary interpretation and embellishment of her powers that were being reenacted, but the actual powers/stats that she had back when she was alive. And the main difference between Black Saber was that she did receive actual flesh from the contaminated Grail as well as an immense load of Mana meaning that she could be greater than what she was back when she was alive, which I'm interpreting to mean...
Black Saber > Arthur = Rin Saber >>> Shirou Saber.

What I'm trying to get at is that in the transition from Heroic Spirit to Servant, information can only be interpreted (aka filtered) using the information that survives in "history" and the basic archetypes provided by the classes. Legends with no actual "historical" truth won't add anything.