These are both correct.

If you inhale through both nostrils, it tends to activity the SNS; exhale through both activates the PNS.

It's different when you close off one nostril, and most research suggests it's as you say - inhaling through left nostril activating right hemisphere (not "brain"; right hemisphere associated with the PNS- for the inevitable skeptics about hemispheric differences, check Iain McGilchrist's videos) and vice versa.

Good point!