Why does $e$ have multiple definitions?
The number $e$ seems to have multiple definitions:
$$\lim \limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$
The unique number $a$ such that $\int_1^a\frac{1}{x} \, dx = 1$
The unique number $a$ such that $\frac{d}{dx}a^x=a^x$
The base of the natural logarithm
This always seems strange to me. Why is that? Why is there no 'agreed upon' definition, and all the other definitions are theorems? Some of the 'definitions' of $e$ are theorems that have to be proved if you use another definition of $e$. This can make looking up the proof for something involving $e$ or $e^x$ confusing and possibly promote circular reasoning.
Analogously, here are several ways to define me:
I am the citizen of the US with social security number [XYZ]. This is of primary interest to the government.
I am the oldest son of [my mother's name]. This is of primary interest to my family.
I am the instructor of [particular course meeting at particular days/times] at [university]. This is of primary interest to students in that class.
I am the author of [a particular master's thesis]. This is (maybe) of primary interest to my thesis advisor.
Of the above list, which is "the right definition" of me?
As you can see, I am related to the world in a multitude of very specific ways. Though some are quite different in their nature, they all determine me uniquely, with different people and institutions thinking of me primarily in different ways.
Similarly, the constant $e$ is related to various pieces of mathematics in many different, but specific, ways. The definition used may vary depending on what role $e$ is fulfilling in a particular context, but they all uniquely determine the same constant and are all important for their own reasons.
The number $e$ has many different characterizations.
The word "characterization" has a precise meaning in mathematics. An exercise in a textbook may say:
(a) Prove that $X$ is enormously purple but not largely purple.
(b) Prove that the property of being enormously purple but not largely purple characterizes $X$.
The student is expected to understand the difference between (a) and (b). To say that the property characterizes $X$ means that $X$ is the only thing that has that property.
The number $e$ has many characterizations besides the ones you mention. So does the golden ratio. So does $\pi$. The concept of a "parabola" can be characterized as (1) a certain conic section; or (2) the locus of point equidistant from the focus and the directrix; or (3) the curve having a certain reflecting property; or (4) the graph of a quadratic polynomial function; or any of a number of other ways.
Which characterization should be taken to be the "definition" depends on context. And unfortunately mathematicians have never collected their thoughts on the precise nature of that dependence in the same way they have on matters of deductive logic.
Answer 1: pedagogy.
A definition is chosen because it fits into the story the author wants to write. An author may find a particular choice of definition fits nicely into their story; e.g. one author may prefer start from the simplicity and elegance of manipulating integrals, another might want to talk about solving differential equations, and yet another author might want to start talking about $e^x$ early on and so prefers the one in terms of limit.
Answer 2: they're 'different' constants, and it's a neat theorem that they all turn out to be equal!