Help formulating node equations
What is wrong with my logic?


Your 3rd equation is not consistent with the passive sign convention.

Since you've chosen \$i_1\$ to enter the \$e_1\$ terminal of \$R_1\$, the correct equation is, by the passive sign convention,

$$i_1 = \frac{e_1 - e_2}{R_1}$$

Note that one does not need to know in advance if \$e_1\$ is a higher potential than \$e_2\$.  It may be that, when the equations are solved, \$e_1 \lt e_2\$.

But that's irrelevant since, in that case, \$i_1\$ will be negative and thus, as desired, the current will be entering the more positive terminal of the resistor (a negative current to the right is a positive current to the left).

Given the reference polarities and reference directions chosen, the correct KCL equation for node 2 is

$$i_1 - i_2 + i_3 = \frac{5 - e_2}{3} - \frac{e_2}{5} + 3 = 0$$

which is equivalent to your 1st equation.