“Cloud storage” (camplus, free cloud, anything in the event tab really) and “local storage” (sd card in camera) are seperate and completely Independant of eachother. Settings from one do not effect the other. You can have one, both or neither at anytime. Think of it as two seperate entities and functions.
Technically there is a third way of recording, the record button you referenced earlier. This saves either the live view or playback (sd card) coverage you are watching or reviewing in real time to your device storage.
Late addition/edit:
Both local storage and cloud storage have their own “event recording” that works each a little different. Camplus records a clip that is for the duration of the motion detection. Motion stops, clip is uploaded to the cloud and is accessable in the event tab. When local storage has event recording enabled and a.motion event is detected, the full minute where motion is detected is saved to the card. If motion is from :25 to :35 seconds on the watch then :00 to :59 will be saved to your card and accessable via the playback button in live view. If motion goes over into the next minute mark, the full next minute will be saved.