Lightspeed is currently going through a transition phase where most On Premise POS is transformed into cloud account. APPSeCONNECT is connecting Lightspeed Retail Cloud POS system, but since On Premise can also be accessed through same customer credentials, so yes APPSeCONNECT can sync data to Lightspeed Retail On Premise version as well.
Our integrations are dynamic in nature. The mentioned offerings are the cumulative possibilities of the platform (actual scenarios are specifically organization/business-dependent). For any disparity, please refer to the latest version of the integration brochures.