If you buy stocks of a company you are buying a share in that business.
If you buy bonds from the same company you are essentially loaning them money for future payment.

I am guessing (correct me if I am wrong!) that if you buy a bond the terms of repayment have to be very concrete, i.e. you get 2 dollars next year and 10 after that. Otherwise they could tag the repayment to the stock price and what you would have is a bond that behaves like a stock with automated dividends.

My question:
What body of law governs what you can and cannot promise in a bond contract and what agency is overseeing this? (Corporate law and the SEC?) I am most interested in the US setting, but info about other countries is also welcome.

A secondary question is what big practical differences between stocks and bonds the regulation causes. (I don't mean economic consequences such as stocks have higher yields but bigger 'risk', but practical differences that lie in the mechanism such as bond owners having precedence in case of bankruptcy.)

My question was inspired by EnergyNumber's comment on this question.