What is/are the difference/s between (1) and (2) - as described in the op?
All I know is \(\displaystyle :=\) means "is defined as" and \(\displaystyle =\) means "is equal to".
So, you're asking whether that formula is the definition, or a result of some other definition?
In natural language we use ":=" to (re)define a (new/old) word/symbol. E.g. from now on "xt" means "the wolf without a tail" i.e. \(\displaystyle \text{xt}:= \text{wolf without a tail}\). Then you know that when I write "xt ate the old man", I mean the wolf without a tail ate the old man. My hunch is that it's a part of a given \(\displaystyle \mathbb{D}\) (domain of discourse), in which you want to use a special word/symbol to mean something specific for as long as we're within \(\displaystyle \mathbb{D}\).So, you're asking whether that formula is the definition, or a result of some other definition?
That depends on what you use as the definition! Did you try looking it up?
Hyperbolic functions - Wikipedia
en.wikipedia.org
You'll see that you can start with various definitions, that formula being one of them. So your question is moot. The answer is, it can be, or not.