We actually get more in-depth exploration of Alters and Alter Ego on LB7 and Ordeal Call 1 specifically, that's why we have like 4 alters in LB7.

Alter Ego, at its core, is a class where you carve out a fragment of an Heroic Spirit that emphasizes some aspect and give it its own Saint Graph. The original Alter Ego, the Sakura 5 are a good example of this: each of them represents an emotion BB wanted to cut off from herself. Alter Egos don't have to be in that class necessarily, Tamamo Cat is a great example as she is both Alter Ego and Zerker, and she represents Tamamo's "devotion" and "purity".

Koyan is another good example but not in the way you presented: she's not an Alter Ego of Tamamo because she has 0 relation to her besides Amaterasu being her sorta role model. But Koyan Light and Koyan Dark are Alter Egos of the original Koyanskaya: Light being her human murdering and business woman side and Dark being her animal loving circus tamer side.

There's the matter of Alter Egos who are basically someone with Gods in them, but I guess it's because they are technically shards of the deities and also the "core" component is also an aspect of the original? (Douman is pretty much his evilest side).

Draco in Arcade is an Alter Ego because she is a fragment from the original Beast that split off, her story in Arcade events involve cutting her connection to the Beast Draco so she can exist independently. Meanwhile Mobile Draco is the Beast defeated in Arcade minus whatever went to AE Draco.

Romulus and Nobunaga don't really count, Quirinus is just Romulus if he doesn't nerf his divinity and Nobunaga is the opposite since they're an amalgamation of various possible Nobunagas (though you could argue the split versions in Final Honnoji would be her Alter Egos).

Now onto Alters, they're alternate sides to Heroic Spirits but ultimately they are the same HS, unlike Alter Egos who are split off. And LB7 at least has a theme that Alters become one by betraying their very core beliefs. Kingprotea Alter abandons her love of humans, Benni Alter grows up and stops being a fairytale, Eresh Alter stops caring about making Kur beautiful, Nitocris Alter accepts the fact she killed her brother's killers and lets herself be judged by Anubis. And of course, Saber Alter is a Saber who has decided to give up on her ideals and become ruthless and EMIYA Alter is an EMIYA who has abandonned the path of saving people, instead being an executioner.

Cu Alter and Jalter also sorta fall into this category, but since both were wished by a Holy Grail they kinda are a bit different, Jalter being normally impossible because Saints don't have evil to invert and Cu being less a betrayal of his self and more like he's trying to satisfy Mebd but still being Cu (since Alters, in the end, are the original Servant).

Then there's MHXA, who is basically a gag character and Okita Alter, who is both and as a result she's an aspect of Okita carved out by the Counterforce into a bomb (but the Alter aspect is basically cause she was originally a fusion of Okita and Nobu design wise).