The extended warranty on my car says "Replacement parts may be new,
remanufactured, or of like kind and quality." What is the legal
meaning of "like kind and quality" in Ontario?

I have seen it defined as "meaning used parts from a salvage vehicle
of the same make and model, usually from the same year or newer than
the vehicle being repaired with comparable or less mileage than the
vehicle it is being used for". I am not sure
(1) whether this definition applies in Ontario
(2) whether this definition applies to the specific phrasing in my
contract (does "like" mean "like new or remanufactured" or "like the
orignal part that broke"?).

The warranty company is offering to replace the part with a used part
out. I haven't been able to find out the year or mileage of the car it
came out of - I don't think the supplier knows. Does this matter?

Telling the Researchers what the part is may help.

Can you negotiate with the warranty/insurance company to have them pay
the dollar amount for the part they will supply and you pay the
difference to get a new one?

The part is an ignition coil.

The warranty company has offered either a used part (mileage unknown)
of $130 toward the cost of a new part ($500 or so). I would like them
to pay for either a new or a remanufactured part, or for a part that
is demonstrable of similar quality to a new or remanufactured part.

Thanks for your comment.