Often times a chat log is needed when people don’t check it often or right away.
I think the timer on each device starts from when the person who has the devices sees the message.
So if you send the message and the timer is 5 minutes, the message on your account (on all of your devices) will be deleted in 5 minutes from now, while the recipient will first see the message (maybe in an hour) and then after 5 minutes it will be deleted from their devices too.