There are several good reasons why I should not assign a probability of 66% to heads and 66% to tails, but one of the clearest is this: you can make me a bet that I will give you $2 if it lands on tails and you give me $1 if it lands on heads, and then a second bet where I give you $2 if it lands on heads and you give me $1 if it lands on tails.

Got it.

Whichever way the coin lands, I owe you $1 and you owe me $2 - I have gained a free dollar.

Huh? You swapped "you" for "I" here (compared to above).