Once it got popular, "Agile" was co-opted by all the usual players (cf "extreme" before it, to a lesser degree).
The important idea is "agile" vs. waterfall, whether or not that includes anything directly recognizable as "Agile". Or call it something else, doesn't really matter.
Recent history shows you can certainly do things directly recognizable as part of the Agile methodology while demonstrably not being agile, so modulo the no true scotsman fallacy it's a much less fruitful distinction to draw.