The pattern I see is that Euclidean space is an ordered pair, first element of which is in some way related to $\mathbb{R}$, sometimes to $\mathbb{Q}$ and $\mathbb{C}$, the second one is a function on it.
But the precise elements vary in different sources - this website alone has two different definitions of the first element: first it says that it's a cartesian space, but right in the next section, it allows it to be instead an ordered tuple - vector space, a module. Encyclopedia of Math adds to the confusion by using the second case, the "module" definition in the definition of the pair forming Euclidean space.
Second element of the definition is a function, which can be either - according to Proofwiki and Wikipedia, Euclidean metric, $\sqrt{\sum_{i \mathop = 1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^2}$ or, as according to Encyclopedia to Math, inner product ${\sum_{i \mathop = 1}^n {{x_i, y_i}}}$ with $x=(x_1,...,x_n)$ and $y=(y_1,...,y_n)$.
Which definition is correct? Surely all of them cannot be, unless definition is different in different branches in mathematics (if that's the case, which branches use which definitions?).