TRIAC turning on asymetrically by itself when it gets warm
I see two issues with what you're trying to achieve:
1. I assume you want to control the speed of the motor by adjusting the firing angle of the triac. AC motors do not respond well to lowering the RMS voltage applied to them which happens when you delay the firing angle. When a motor is operated at less than its rated voltage, it compensates by drawing more current to maintain its mechanical power output. I'm not surprised the motor burns out. If you want to control the speed of the motor, you need to use a variable speed drive which varies the frequency of AC to the motor.
2. The graph shows what looks like a spike at the leading edge of each waveform. If you expand the timebase, you'll likely see it better. Check its amplitude. The triac heats up because it's switching on/off every cycle. The more frequent the triac switches, the hotter it gets. It is during the switching transient from off to on or on to off that the triac consumes power, albeit for a short interval of time. When it fully on or fully off, it dissipates very little power. During these switching states from off to on, the motor delivers a voltage spike that likely exceeds the forward voltage of the triac and causes it to turn on despite the snubber network. If you exceed a triacs forward voltage it will turn on even without gate drive. The collapsing magnetic circuit of the motor is generating a voltage spike that retriggers the triac. I suspect the threshold lowers with rising temperature. I think the circuit will work OK for simply turning on or off the motor for long periods of time (when the triac's not constantly switching) but don't try using it to change motor speed by varying phase angle.