I think the third condition is false conditional on Schanuel’s conjecture. 

For example, we can use Schanuel’s conjecture three times to rule out the possibility that $e$ is the solution to $2^{2^u}-u=93$. I hope and expect that this (or some simplification) provides a template for ruling out all possible existential definitions of $e$.

If the above equation holds with $u=e$, then consider the four pairs of expressions:

$$w=\ln 2, \           e^w=2\\
x=1,   \               e^x=e\\
y=e \ln 2,   \      e^y=2^e\\
z=(2^e) \ln 2,\  e^z=2^{2^e}$$
 
We have five independent algebraic relationships among them:

$$e^z - e^x = 93\\
e^w = 2\\
x = 1\\
y = w e^x\\
z = w e^y$$

By Schanuel’s conjecture (in a [version][1] that looks natural constructively, and can be considered a contrapositive of the usual [statement][2]) there must be a rational linear dependence among the first four variables:

$$a w + b x + c y + d z = 0$$

So the three pairs of expressions with $w,x,y$ have four independent algebraic relationships between them, where the first comes from dividing the last equation by $\ln 2$:

$$a + b/w + c e^x + d e^y = 0\\
e^w = 2\\
x = 1\\
y = w e^x$$

By Schanuel’s conjecture there must be a rational linear dependence:

$$p w + q x + r y = 0$$

So the two pairs of expressions with $w,x$ have three independent algebraic relationships, where the first comes from dividing the last equation by $\ln 2$:

$$p + q/w + r e^x = 0\\
e^w = 2\\
x = 1$$

By Schanuel’s conjecture there must be a rational linear dependence

$$s w + t x = 0$$

This would mean $\ln 2$ is rational, which is impossible, so the original equation does not hold for $u=e$.


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