Connecting the two definitions of $e$

Let's start with your second definition. We want to find the derivative of $f(x) = a^x$, for some $a>0.$ Using the definition of derivative, we get

\begin{align} f'(x) &= \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}\\ &= \lim_{h\to 0}\frac{a^{x+h} - a^{x}}{h}\\ &=a^{x}\lim_{h\to 0}\frac{a^{h} - 1}{h}. \end{align}

Now, as you said, one way to define $e$ is to define it such that $$\lim_{h\to 0}\frac{e^{h} - 1}{h} = 1.\tag{$*$}$$ Now, I'm going to be somewhat informal here. Suppose we have $$\frac{e^{h} - 1}{h} = 1.$$ Then, doing some manipulation: \begin{align} \frac{e^{h} - 1}{h} &= 1\\ e^{h} - 1 &= h\\ e^{h} &= h + 1\\ e &= (h+1)^{1/h} \end{align}

Now, by taking limits we have that \begin{equation*} \lim_{h\to 0}\frac{e^{h} - 1}{h} = 1 \iff e = \lim_{h\to 0}(h+1)^{1/h}. \end{equation*}

Taking the new limit, $e = \lim_{h\to 0}(h+1)^{1/h}$, and setting $n = 1/h$, we note that $h\to 0 \implies n\to \infty$, and we can change variables to get $$e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n},$$ which matches your first definition.

As to understanding where the natural log comes from, i.e. why we have $$\lim_{h\to 0}\frac{a^{h}-1}{h} = \ln(a),$$ this is a straightforward consequence of $(*)$: \begin{align} \lim_{h\to 0}\frac{a^{h} - 1}{h} &= \lim_{h\to 0}\frac{e^{\ln(a^{h})}-1}{h}\\ &=\lim_{h\to 0}\frac{e^{h\ln(a)}-1}{h}\\ &=\frac{\ln(a)}{\ln(a)}\lim_{h\to 0}\frac{e^{h\ln(a)}-1}{h}\\ &=\ln(a)\lim_{h\to 0}\frac{e^{h\ln(a)}-1}{h\ln(a)}.\\ \end{align} Now, set $k = h\ln(a)$, and note that because $a > 0$, we have that $h\to 0\implies k \to 0$ as well, and so we have by $(*)$ that: \begin{align} \lim_{h\to 0}\frac{a^{h} - 1}{h} &=\ln(a)\lim_{h\to 0}\frac{e^{h\ln(a)}-1}{h\ln(a)}\\ &=\ln(a)\lim_{k\to 0}\frac{e^{k} - 1}{k}\\ &=\ln(a)\cdot 1\\ &=\ln(a). \end{align}

This Wikipedia article: Characterizations of the exponential function, shows six ways to define the exponential function $e^x$ (thus for the constant $e$) and a detailed discussion of equivalency of all the six definitions.

This happens a lot in mathematics. The same mathematical object may have many properties that can characterize the object. Once they are proved to be equivalent to the definition, they can all be regarded as a definition. More generally, there are equivalent definitions of mathematical structures.

For an intuitive understanding of the constant $e$ and exponential growth, you may read this article https://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/ from the "BetterExplained" website.