You'll know when the client is right since it will say Heavensward and have different start screen music logo.

I only play on PC and the PC client automatically updates once the game is registered. I don't know how Sony handles it on their consoles. I imagine that, since you have the client installed (it changed your home screen background), it's probably there and you might have to manually find it and select it.

Maybe someone who knows the console hoops better can direct you.