About the first part of your question:
There is no need for an 'agreed upon' definition, if you have the given scenario.
You can choose any definition to start from and then prove your way through to the theorems that state equality to the other ones.
For instance, take two persons, person 1 (p1) and person 2 (p2).
Furthermore assume that p1 defines $e$ via definition A, and that p2 defines $e$ via definition B.
Then p1 can prove a theorem that states that $e$, defined via definition A, is equal to $e$, defined via definition B/ has the defining and unique property used in definition B.
So p1 can start proving the same theorems about 'his' $e$ (defined as in definition A) as p2 about 'his' $e$ (defined via B) and vice versa.
So, while strictly formally speaking, p1 and p2 are talking about different $e$'s, from an epistemological point of view, it doesn't make a difference:
Any theorems using properties of (any) $e$ hold true in both "branches" of mathematics that p1 and p2 are building by using different definitions (assuming that up to chosing different definitions of $e$, p1 and p2 went through mathematics with the just the same doesn't harm my argument here).
About the second part of your question:
Strictly speaking, you can only define a term once on your way through mathematics. Defining is simply giving initial meaning to something:
In mathematics, a definition is used to give a precise meaning to a new term, instead of describing a pre-existing term.
(quoted from Wikipedia with the liberty of making the word 'new' bold)
So to put it simple... since there can be no other definition of $e$ besides the one you have opted for on your educational/epistemological path through mathematics, the other optional definitions you could initially chose from can at most become statements about 'your' $e$ that have to be proven - and if you have proven them, they have become theorems/corollaries etc. about 'your' $e$.