However, very early in the movie he says "If you make yourself more than just a man, if you devote yourself to an ideal, you become something else entirely" so it seems obvious in hindsight that this applies to Ra's as well as if applies to Batman.

But that doesn't mean Henri Ducard became a Ra's Al Ghul or Ra's Al Ghul became the League of Shadows. It's like a religious thing, like christians consider themselves reborn after baptism.

Hell, I always thought Ken Watanabe was Ra's Al Ghul and the title passed on to Liam Neeson after the former died.

I can see where you're coming from, I'd expect some quote like "I am Ra's Al Ghul now" or something.

Again, I can see your interpretation, but I think Ra's Al Ghul simply being a name is far less a stretch and requires fewer assumptions.