So, the chain contains a full history of transactions, and a coin's current owner is determined by all nodes as being the last signature in the chain associated with a coin. Am I reading this right?
Yes. Whoever has the private key that can create that last signature can spend the coin-- he or she (or them-- eventually maybe as bitcoin clients add features) are the owner.The transaction history isn't exactly a "chain" -- multiple coins can get combined as input to a transaction (they're all "spent"), and multiple coins can get produced from a transaction (they're all "unspent" until they're used as the input to another transaction), so it's a more complicated network (I think of a chain as one link followed by another in a straight line). But it will all trace back to one or more 50 Bitcoin GENERATE transactions.