‘The lights changed’ or ‘the light turned green’ sound most natural, I think. Otherwise, ‘went green/red’ is better than ‘get.’ ‘Go’ seems to indicate a faster, less gradual change than ‘get’. ‘Get’ may apply when the color starts in one area and spreads to the whole. For people/faces changing color, either go or get would work. Some examples:
‘When the light turned green, she accelerated.’
‘When the light went green, the car stalled.’
‘The lights changed and she turned right into the side street.’
‘His face got red when he tried to lift the heavy weight.’
‘He went red when she gave him a peck on the cheek.’
About the second situation, it’s worth noting that in the US, the legality of being in the intersection when the light turns red actually differs by state. See the second paragraph in the link below.
This source uses the technical jargon of 'traffic engineers'. Traffic police and others might also use this kind of language ('failing to stop' instead of 'not stopping' for instance.) However, describing what happens in such precise detail would be unusual in everyday speech.
But when the light is red and you go anyway, that’s called 'running a red light'.