I think the third condition is false conditional on Schanuel’s conjecture. The algorithm below handles most potential examples, with a bit of uncertainty only about to handle extra algebraic dependencies.

We can use Schanuel’s conjecture to turn potential algebraic relationships into linear relationships, by a [version][1] that looks natural constructively, and can be considered a contrapositive of the usual [statement][2].

As an example, we have a preliminary lemma: There is no linear dependence of the form $a\ln 2 + b + c e\ln 2=0$ with rational $a,b,c$. Proof of lemma: We consider the pair of expressions $(u=\ln 2, e^u=2)$, $(v=1,e^v=e)$, and then have three algebraic relations among them: $e^u=2, v=1, au+b+ce^vu=0$. By Schanuel's conjecture, we must have a rational linear dependence of the form $p \ln 2+q\,1=0$, which is impossible.

Now suppose we have a definition of $e$ as suggested in the post, with $m$ equations (that may involve polynomials and powers of two) in $m$ variables $x_1,\ldots,x_m$, jointly implying $x_1=e$. Wherever those equations include a term of the form $2^t$, we replace $t$ with $x_{m+1}$, and add the equation $x_{m+1}=t$; this gives us $m+1$ equations in $m$ unknowns. We continue this until the only powers of two which appear are of the form $2^{x_i}$, and then have $n$ algebraic equations in the $n$ variables $x_i$ and the corresponding terms $2^{x_i}$. We can assume wlog that $n$ is minimized and that the equations are of minimal complexity.

To use Schanuel's conjecture, we consider the $n+2$ pairs of expressions $(u=\ln 2,e^u=2)$, $(v=1, e^v=e)$, $(w_i=x_i \ln 2, e^{w_i}=2^{x_i})$. We can rewrite the $n$ equations above in terms of those expressions, replacing $2^{x_i}$ by $e^{w_i}$ and replacing other instances of $x_i$ by $w_i/u$. This gives $n$ algebraic relationships among the $n+2$ pairs of expressions, and then we add $e^u=2$, $v=1$, $w_1/u=e^v$ to get a total of $n+3$ algebraic relationships.

If these are $n+3$ independent algebraic relationships, then Schanuel's conjecture gives us a rational linear dependence of the form
$$a u + b v + \sum c_i w_i=0$$ 
By the lemma, some $c_i$ must be non-zero with $i>1$. So for that $i$, so we can further transform the equations by
$$w_i \to \frac{-1}{c_i}(a u + b v + \sum_{j\neq i}c_j w_j)$$
$$e^{w_i} \to \Big((e^u)^a (e^v)^b \prod_{j\neq i}(e^{w_j})^{c_j}\Big)^{-1/c_i}$$ 
This eliminates $x_i$ and gives $n+2$ algebraic equations in $n+1$ pairs of variables. We continue using Schanuel's conjecture to eliminate variables until we get $4$ equations in $3$ variables, and a linear dependence as in the lemma. Since this is impossible, it disproves the hypothesis that the original equations implied $x_1=e$.

Alternatively, at some point we might find that the $n+3$ algebraic relationships are not independent. In this case (the only part where I haven't figured out how to write things down in detail) we can find solutions to the equations with $x_1\neq e$, again disproving the initial hypothesis.

For example, we can carry out this process for

- an attempted definition of $e$ by itself: $2^{2^x}-x=93$
- an attempted joint definition of $e$ and $\pi$: $2^x-y/2=5$, $2^y x = 24$
- an attempted definition of $e$ that is too weak, and therefore leads to algebraic dependencies: $2^{x+3}-8 \cdot 2^x=0$

Previous versions of the post give some details for those cases.

  [1]: https://mathoverflow.net/a/163965/44143
  [2]: https://en.m.wikipedia.org/wiki/Schanuel's_conjecture