I think there are two issues here. First, "If we get into an accident, am I not covered?" Like Cap said, "you would have to prove a relationship to be covered". Being covered would probably not happen if you were not related as insurance companies look for any loophole not to cover. You're being a visitor would probably disqualify you immediately.
Secondly, I travel frequently in white plated cars where I live in the interior but I do it at my own risk. There are few Red Plates in rural Jamaica especially in sparsely populated areas. A lot of the time I am not paying taxi fares either. If someone is going to Santa (the town near where I live), I tag along and chip in for the petrol. I am under no illusions that I am safe but my friends do drive cautiously or I don't ride with them.
As Rob says, "Use a Red Plate Taxi". It is the best policy. I wouldn't be able to go anywhere around where I live if I had to always use one, however.
Peace and Guidance