What is the stability of a digital signal processed by a filter with z-transform?
The mistake was here: -

You assumed that the rules that apply on the s-plane also apply to the z-plane. They don't because when you map the left hand side of the s-plane to the z-plane you get a unit circle: -

It's a unit circle (amplitude 1) because the s-plane Nyquist frequency is \$\pi\$ radians per second (0.5 Hz). So, everything inside the rectangle on the left side of the s-plane bounded by the top horizontal magenta/purple line down to the lower horizontal dotted black line is within or on the unit circle in the z-plane.
So, given that you calculated z-plane pole values of -1 and -2, the -2 pole is clearly unstable.