Hi, no problem! I was going around the area resolving simple notes. But since this one was not so simple, I thought it was worth commenting.

I agree with your first takeaway, but I don't interpret the second quote the same way. I do think the phrasing is misleading though, so I see where you're coming from.

The difference between railway=station and railway=stop is that railway=stop is a node on the way a train route travels on, representing where the vehicle actually stops (like https://www.openstreetmap.org/node/6217647183). Meanwhile, railway=station represents the conceptual unified station (like https://www.openstreetmap.org/node/6215060982), which can have multiple stops be part of it. railway=station is not usually tagged as attached to the tracks themselves since one node can apply to multiple tracks. I think we agree so far.

As far as the 'stop' role, note how https://wiki.openstreetmap.org/wiki/Relation:route#Members says it's for a node "on the route road way". Since railway=station and railway=halt are not tagged on the route road way (that is, attached to the tracks), to me this implies that the role should be applied to the relevant railway=stop, which is attached to the tracks. In that case, the author of that line of the wiki was just using "train halt/station" colloquially to mean where the train stops, which ended up being a bit misleading in context.

I note that in a few other places I spot checked, similar railway=stop nodes are tagged with the role 'stop' in the subway route relations, such as in San Francisco (https://www.openstreetmap.org/node/2208932570), New York (https://www.openstreetmap.org/node/1695334044), and Munich (https://www.openstreetmap.org/node/2622285804). That's anecdotal to be sure, but it indicates to me that the existing mapping is at the very least common, and perhaps even the predominant way to tag this worldwide.