They kept records back then ?
Yep. The exact date is from Japanese tsunami records.
The back up data is entire drowned forests whose last growth season, by tree rings, was 1699.
The Japanese government did. They got walloped with a huge tsunami and recorded quite a bit of information about it. It was in more recent times that American geologists compared notes with the Japanese.
They have very accurate records of that quake to the day and hour it happened. The Japanese recorded the time of the tele tsunami when it hit them. Scientist just back tracked to how long it would take to go transpacific. Geological studies showed the results of the near tsunami in Washington State.
Japanese record keeping noted a tsunami without a preceding earthquake—a bolt out of the blue kind of wake-up.