Understanding notation and meaning of uniform convergence of a power series

I am trying to understand correctly the idea of uniform convergence in power series, and in general the notation for series of functions.

When we define a series of functions, we are defining a ‘new’ object, in which the terms of the series are functions rather than real values. Then we define this new object’s value in terms of limits of sequences of functions. All of this is analogous to defining what a series of real values if, and we write:
$$sum_{n=1}^{infty}f_n = f$$

and we specify whether we mean uniformly or pointwise.

Here the series is a summation of functions, and its value is precisely a function. This seems to be quite commonly written:

$$sum_{n=1}^{infty}f_n(x) = f(x)$$

taken to mean the exact same thing as the previous equality (including whether it is pointwise or uniformly convergent to $f$). This notation sometimes becomes a bit confusing for me, because dependent on context we either mean $f(x)$ is a function itself, or $f(x)$ is the real value image of some parameter $x$ under $f$.

Say I define a power series and write $sum_{n=1}^{infty}a_n x^n = f(x)$ In this case, the symbol $x$ refers to a parameter, and $f(x)$ is simply the limit of the series of real values whenever $x$ lies in the radius of convergence. This means that the power series IS the function $f$, across the radius of convergence.

My question is then to clarify the exact meaning and notation when we say a power series is uniformly convergent on an interval. We have that $sum_{n=1}^{infty}a_n x^n = f(x)$ (in defining the power series), and I have often seen the uniform convergence written as $sum_{n=1}^{infty}a_n x^n = f(x)$ (uniformly), where sometimes the ‘(uniformly)’ is dropped. I am hoping to clarify my understanding of these two identical notations.

Am I correct in saying that the first notation (defining the power series as a function), is information about the convergence of real valued series within the radius of convergence? That is I treat $x$ as some particular but unspecified value, and so the series is NOT a series of functions, but of real values? I.e. this allows me to create a well defined function $f$, whose domain is the radius of convergence.

In the second notation, is it correct to say that now $f_n(x)$ is being treated not as a real value on some parameter $x$, but instead to say that it IS a function? I.e. $f_n(x) = f_n$, where $f_n$ is defined by $f_n(x) = a_n x^n$? That is to say, the second notation asserts that the series of functions $sum_{n=1}^{infty}f_n$ is uniformly convergent to the function $f$ (which was defined above).

Ultimately, is asserting that a ‘power series is uniformly convergent’ equivalent to saying that a series of functions (given by $sum_{n=1}^{infty}f_n$) uniformly converges onto the power series itself?