You also mention Holy Warriors but to me that is more paladin than cleric.
Yeah it's different eras of D&D. In 2E, Cleric is
explicitly called out as a "Holy Warrior" and specifically compared to holy knightly orders* who I think later D&D would more closely associate with Paladins. In earlier D&D, Paladins definitely aren't "Holy Warriors" in the same sense at all, rather the Paladin deal is 100% about being "noble and honourable" and following this exacting code of honour above all else, and being Lawful Good is a big part of that. If you read the 2E Paladin description, no mention at all is made of them following a deity, and whilst they do have to seek out a Cleric to atone if they do something Chaotic, the requirement isn't that the Cleric has the same deity or the like - it's that the Cleric be Lawful Good. So there's an implication of faith (also all the literary examples referred to in the entry are famously Christian knights*), but not a requirement.
Even in the Complete Paladin's Handbook in 2E, it's stressed in chapter 8, "Faith", that not all Paladins follow a specific religion, let alone a specific god, and that some follow philosophies or the slightly mysterious "non-traditional philosophies" or "natural forces". The key is that they have to be LG and follow a strict code of conduct.
Over the course of 2E (and people had been thinking about this since 1E or earlier), people increasingly started thinking about non-LG Paladins, not least because a huge proportion (a large minority at least) of 2E Paladins ended up being RP'd as "Lawful Stupid" and generally a pain in the behind.
In 3E the LG restriction remained initially, continuing to prevent Paladins from really being "Holy Warriors" (because only a minority of gods would be able to have such, which would make no sense) and indeed says Paladins don't even need to worship a specific god explicitly, just to be LG. However a lot of people were now thinking Paladins did have to have a specific deity, an idea that the Forgotten Realms had promoted in 2E. And 3PP stuff quickly came out supporting other alignments and then we got official versions in 2004's Unearthed Arcana.
And that's where the changeover really happens, where the Paladin starts really occupying the "Holy Warrior" archetype. By the time 4E comes out, it's formalized. Paladin is in a Defender role for the Divine power source, so very literally and undeniably a Holy Warrior. 4E Paladin also has more flashy and magical abilities than earlier Paladins, something which continues with 5E Paladins.
* = Interestingly these are different selections. The Cleric is compared to medieval semi-monastic holy orders of Knights who theoretically served the Church (Templars, Hospitallers), and also to the Sohei of Japan. People who are told what to do by religious leaders and motivated primarily by religion. The Paladin is compared primarily to individual knights famous for their good deeds and heroism, and nobility - Roland, Lancelot, Galahad, etc. - people who weren't in service of the Church, who didn't have religious leaders giving them orders, but who were just so intensely noble and virtuous that God favoured them. It's kind of like the other way around - Cleric works for a specific god, and does what he's told. Paladin is such a good dude that despite following a code, not orders, they somehow have divine-seeming protection and powers. The Complete Handbook clarifies the latter into it being the Paladins
own faith, not and outside force, grants them these powers/protections.
