How can I analyze an op-amp based negative impedance converter in a circuit?
I think, the equations as given by you are correct. There is no contradiction.
Both currents (\$I_0\$ and \$I_1\$) are going into the circuit and into the opamp's output.
Where is the problem?

EDIT: Here are the formulas, replacing \$r\$ with \$R\$ and \$S\$ with \$R_s\$:

$$I_1 = \frac{V_{-}-V_{1}}{R} = \frac{V_{out}-V_{-}}{R_s}$$

$$I_0 = \frac{V_0 - V_{out}}{R_s} = -\frac{V_{out} - V_+}{R_s} = -\frac{V_{out} - V_-}{R_s}$$

so \$I_0 = -I_1\$, and

$$V_0 - V_1 = I_1R = -I_0R$$