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

Here is an example of using Schanuel’s conjecture 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 $e$.

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:

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

By Schanuel’s conjecture there must be a rational linear dependence among the first four variables:

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

and dividing by $\ln 2$ gives

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

So the three pairs of expressions with $w,x,y$ have four independent algebraic relationships between them:

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

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

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

and dividing by $\ln 2$ gives

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

So the two pairs of expressions with $w,x$ have three independent algebraic relationships:

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

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

$$s w + t x = 0$$

which means $\ln 2$ is rational, which is impossible.