Is there a difference in the meaning of Taylor Series and Taylor Series Expansion? For example:
The Taylor Series of the exponential function about $0$ is: $$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$
Whereas the Taylor Series Expansion about $0$ is: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
Is this correct? I don't mean to be overly pedantic, but I am interested in knowing if there is actually a difference in their definitions. Thank you!
They are the same. The Taylor series is an expansion of a function into an infinite sum.
Both the Sigma notation and the pattern form you listed are equivalent—the summation form is somewhat more rigorous when it comes to notation as it provides a precise definition for the infinite series, while the second way you wrote is a way to quickly grasp intuitively what the series looks like. But it has nothing to do with one being an expansion and the other not.
The goal is for a person reading to be able to grasp precisely how one constructs the terms in the infinite series, so authors will use the representations interchangeably depending on the context.