You should be contracting your trunk muscles when you take your bracing breath before you initiate a lift, but you shouldn’t be concerned with actively trying to achieve lumbar extension when doing so in any of the exercises you mentioned. There’s an increasing amount of research, which Feigenbaum, Baraki, Miles, Ray, etc. have talked a bunch about, that suggests that lumbar extension and flexion don’t necessarily increase or reduce lower back injury risk. You’d be better off focusing on achieving a nicely braced and stacked torso than a lumbar curve.
The only exercise that you probably should try to extend/curve your lower back for is bench press, for the sake of making a better arch. But even then, if you wanted to bench with your back completely flat on the bench for a longer ROM, I suppose that would be fine for general strength purposes.