I suspect you both are looking at it from two different perspectives.
You are looking at it it seems from the perspective of what the PC looks like on battlefield. And yes, in that way a war cleric and a knowledge cleric are wearing different things, are wielding different weapons, etc. They don't present as the same type of character under the same class.
But I suspect from
@Maxperson 's perspective... they're thinking that all Clerics are the same narratively in that they are all divine acolytes of a specific god, each of which does the same thing and has the same categories of power-- the only differences between them is which gods they serve (and the domain that gods reside over.) As opposed to say clerics that are all different types of priests with different duties or functions-- Missionaries that go out proselytizing, Archivists that collect religious knowledge, Templars that fight on the behalf of their churches etc.
Now sure... obviously we can say that Knowledge Clerics are like Archivists, and War Clerics are like Templars... but the difference would be that EVERY domain could have Archivists and every domain could have Missionaries, and every domain could have Templars. After all... why shouldn't the priesthood of a Knowledge god have Templars who fight with weapons on the god's behalf? Or why shouldn't war gods have those priests back at their temple collecting and storing all the god's important items? And that's where changing up the subclass type would come into play. Warlocks have two different things that change them-- WHO they get their power from (their Patron), and HOW do they use that power (their Pact). Clerics could certainly have the same sort of differential.
Maybe I'm wrong with what you both are talking about... but this was my impression of the conversation and the differences between the two of you.