They're just different from what Lamport originally proposed. Asynchrony as given is roughly equivalent to Lamport's characterization of distributed systems as partially ordered, where some pairs of events can't be said to have occurred before or after one another.

One issue with the definition for concurrency given in the article would seem to be that no concurrent systems can deadlock, since as defined all concurrent systems can progress tasks. Lamport uses the word concurrency for something else: "Two events are concurrent if neither can causally affect the other."

Probably the notion of (a)causality is what the author was alluding to in the "Two files" example: saving two files where order does not matter. If the code had instead been "save file A; read contents of file A;" then, similarly to the client connect/server accept example, the "save" statement and the "read" statement would not be concurrent under Lamport's terminology, as the "save" causally affects the "read."

It's just that the causal relationship between two tasks is a different concept than how those tasks are composed together in a software model, which is a different concept from how those tasks are physically orchestrated on bare metal, and also different from the ordering of events..