Meaning of Exponential map

I've been studying differential geometry using Do Carmo's book. There's the notion of exponential map, but I don't understand why it is called "exponential" map. How does it has something to do with our common notion of exponentiation?

I read from the book The road to reality (by R. Penrose) that it is related to taking exponentiation when making (finite) Lie group elements from Lie algebra elements. It seems like using Taylor's theorem on a manifold so we have, for example, there was the following equation explaining why it is the case.

$f(t) = f(0) + f'(0)t + \frac{1}{2!}f''(0) t^2+\cdots = (1+t\frac{d}{dx}+\frac{1}{2!}t^2\frac{d^2}{dx^2}+\cdots)f(x)|_{x=0} = e^{t\frac{d}{dx}}f(x)|_{x=0}$.

The differential operator can be thought of as a vector field on a manifold, and it is how Lie algebra elements (which are vectors, on a group manifold (Lie group), in a tangent space at the identity element). If I understood correctly, the truth is that this is exactly the exponential map that sends a vector on a tangent space into the manifold in such a way that it becomes the end point of a geodesic (determined by the vector) having the same length.

Why is the above Taylor expansion valid on a manifold? Why is the exponential map the same as taking exponentiation?

The idea of an exponential is the continuous compounding of small actions. Suppose you start with an object $p$, perform an action on it $v$, and then add the result back to the original object. What happens if you instead take half as much action but do it twice? What about if you take one tenth the action but do it ten times? The exponential function tries to capture this idea: $$\exp (\text{action}) = \lim_{n \rightarrow \infty} \left(\text{identity} + \frac{\text{action}}{n}\right)^n.$$

On a differentiable manifold there is no addition, but we can consider this action as pushing a point a short distance in the direction of the tangent vector, $$``\left(\text{identity} + \frac{\text{v}}{n}\right)"p := \text{push }p\text{ by} \frac{1}{n} \text{ units of distance in the }v \text{ direction}.$$

Doing this over and over, we have $``\left(\text{identity} + \frac{\text{v}}{n}\right)^n"p$ means push $p$ by $\frac{1}{n}$ units of distance in the $v$ direction, then push it again in the same direction you already pushed it, and keep doing so until you have pushed it $n$ times.

So long as $\frac{1}{n}$ is small enough that pushing points and vectors in a tangent direction makes sense, what we end out doing is pushing the point $p$ a total of $1$ unit of distance along the geodesic generated by $v$.

Consider the following:

Let $a\in\mathbb{R}$, and define a smooth vector field $X$ on $\mathbb{R}_+$, by $X(p)=ap$. Note that $X$ is left-invariant, when thinking of $\mathbb{R}_+$ as a Lie group. Now let $\gamma$ be a trajectory of $X$, with $\gamma(0)=1$. By solving a simple ODE, one verifies that $\gamma(1)=e^a$, thus the term "exponential map" for Lie gropus coincides with what we know as exponential map on $\mathbb{R}$.