Paganism is incredibly diverse, I will give you that. "Pagan" is more of an umbrella term for many different beliefs with some common elements.
But christianity for example is also an umbrella term - you have catholicism (whis then has the many different orders and stuff under it), evanjelical christianity (with its many denominations) and orthodox church (which may or may not have different groups under it, I don't really know). And even two different people within one denomination of the larger group of christianity may hold a slightly different set of beliefs.
Paganism is just a larger umbrella. I also went with Celtic paganism as it narrows it down a little more, that's why I went for that rather than simply saying "paganism".
Now I get what the hospital tried to go for. But saying it is not tied to a religion is I think a little unfortunate.