Here is a notation for Maclaurin expansions that I made up:
\begin{align*} sin(x) &= \sum \{alt(+); \frac{x^n}{n!}; \text{ n odd $\in$ $\mathbb{N}_0$} \} \\ cos(x) &= \sum \{alt(+); \frac{x^n}{n!}; \text{ n even $\in$ $\mathbb{N}_0$} \} \\ e^x &= \sum \{+; \frac{x^n}{n!}; \text{ n $\in$ $\mathbb{N}_0$} \}\\ \end{align*}
Could this notation be useful?
The $alt(+)$ is indicative of an alternating sign that starts at $+$. Similarly we could have $alt(-)$ that starts at $-$.
Edit:
I have come up with the following notation that might make it easier to read from left to right, that is a bit more compact, and which looks a bit more like the notation that is already in use:
\begin{align*} sin(x) &= \sum_{\text{odd } n = 1}^{\infty}alt(+) \frac{x^n}{n!} \\ cos(x) &= \sum_{\text{even } \space n = 0}^{\infty}alt(+) \frac{x^n}{n!} \\ e^x &= \sum_{\forall \space n = 0}^{\infty}\frac{x^n}{n!}\\ \end{align*}
Here are some properties that I thought might be useful:
\begin{align*} alt(+)alt(+) = +\\ alt(+)alt(-) = - \\ \end{align*}
\begin{align*} e^x = -alt(-)(sin(x) + cos(x)) \\ \end{align*}
This might be useful in the following way:
\begin{align*} sin(jx) &= j.alt(+)sin(x) \\ cos(jx) &= alt(+)cos(x) \\ \\ e^{jx} &= -alt(-)(sin(jx) + cos(jx)) \\ e^{jx} &= -alt(-)(j.alt(+)sin(x) + alt(+)cos(x)) \\ e^{jx} &= -alt(-)alt(+)(jsin(x) + cos(x)) \\ e^{jx} &= -(-1)(jsin(x) + cos(x)) \\ e^{jx} &= jsin(x) + cos(x) \\ e^{jx} &= cos(x) + jsin(x) \\ \end{align*}
You've asked several different questions about the history and purpose of particular mathematical notations. They have different answers.
For single symbols used essentially as adjectives, like $f^*$, $\hat f$ and $f'$ the meaning depends on the context, and it should. There are not enough of them to cover all the possible different uses where a short annotation is really useful.
In your second example you are pointing out one of the shortcomings of the $\Sigma$ notation for sums. It is indeed sometimes hard to unpack. I almost always write my sums with ellipses: $$ \sin(x) = x - \frac{x^2}{2} + \frac{x^3}{3 \times 2} - \cdots $$ with as many terms as I need to see the pattern.
Your two descriptions of integrals are not those of Leibniz and Newton respectively. The first is (essentially) Leibniz. The second is a formal definition of the integral as a limit of Riemann sums. Both are necessary and useful - the first when you are using integrals in a problem in math or physics, the second when you are actually defining integrals and proving their properties.
will finish later ...