I don't want to argue with you but I have garage door sensors and they do not work as you describe. Mine sense distance from ceiling to object. They determine opening and closing (and car presence) based on distance threshhold state changes. Manually opening the door with the rope is no different than any other operation. The sensors can close a dry contact on the opener to trigger open/close.I can definitively answer this with... it depends.
I have had three of these systems. In two of them, there was a sensor on the door itself which detected opening (on my current one, it's a magnetic switch. On my other one, it was a tilt sensor on the door). In one, the open/close status is controlled by the position of the drive mechanism of the opener, so if that was disengaged from the rope, it would not record the open. Upon re-reading, edited to add clarity: on the two systems with a sensor on the door, it absolutely did record the opening even if the rope was pulled and it was opened completely manually. On the third system which sensed it from the opener itself, manually opening the door by hand did not trigger a notification or log anything.
In none of these systems would the open be recorded if power was out. Also I have never seen one of these things give any obvious indication or notification of a restart. And there is no clock in the opener to reset (I think I saw someone mention that). Any timestamps for notifications are from the time on the server, not the opener.
There is no outside server involved. The timestamps are based on the sensor's own clock, which can be altered but by default (in my case) is set by ntp.
Mine are on a small UPS so they do record during power outages, at least for a while. I doubt an 84 year-old woman had this setup but the son or son-in-law could be very tech savvy.
I don't think it makes much sense to speculate on the garage door openings/closings except to note that the time seems to corroborate what the sherrif said about when the son-in-law dropped her off.