Writing a simple sum with einstein notation

The Einstein summation notation includes repeated indices (i.e. summation notation). For vectors (and tensors), $X$ would usually mean the full object, $X = X^{i} e_{i} = X^{0}e_{0} + X^{1}e_{1}...$ etc. The $X^i=X^i (x)$ are the components of the vector and $e_i$ is some basis. The $i$ deonotes the components, and this is summed in the above equation. But we can talk just about the components $X^i$, which means $(X^0, X^1, ... X^n)$.

So yes, you can write $X^i$, but it doesn't mean what you've written down. This is just four-vector notation, but not specifically Einstein summation notation. The sum you wrote is more like the Euclidean norm, which as the other comment says, it's easiest notated as $sum_{i}^{n} X^i $.

Such a sum is, as far as I know, not easily expressible in Einstein notation - you could probably write it in a convoluted way, but that would not really be simpler or easier to understand than $X = sum_i X_i$.

The summation convention is not a shorthand for any sum, but only for certain kinds of sums which arise in linear algebra, such as scalar products, matrix multiplication and so on. In formal terms, the sums at hand are those corresponding to the application of (multi)linear maps between a vector space, its dual, and powers of these; or to the coordinate representation of vectors and tensors. Some examples are given in the Wikipedia article for the convention.

This is the reason for the requirement of two equal indices.