Firstly the quotes you stated are accurate.
Secondly, most people think of a price as amount of cash that you exchange for a thing. However, a futures contract is a contract. It is a legal agreement. The 'price' is just a number in that legal contract.
When you exercise an option on a futures contract, you enter into a contract where the 'price' is the same as the strike price of the option contract.
My interpretation is that the difference between a futures option and
a stock option is that the underlying asset now becomes the futures
contract, instead of the stock
To be precise, the underlying is the futures contract, or rather more precisely, a futures contract that will be entered into if the option contract is exercised (ie, the futures contract details are known when the option is created / sold).
Futures contracts require asset flows from party to exchange (or vice versa) at specific times. The requirements are initial margin (cash), then variation margin as the prices changes daily (cash), then potentially final settlement (cash or commodity depending on the contract specifications set by the exchange).
So you could say that the contract is initially 'free', because there is no initial upfront cash cost to 'buy' a futures contract, but its sloppy nomenclature. You enter into one (long or short) at a given price and then post/receive margin as required.
For the specific example of entering into a futures contract as a result of exercising an option contract, the 'price' in that futures contract will be the strike price of the options contract. This will most likely not be the current market price, and then the futures contract position can be exited by buying or selling the same amount (if it was a put or a call respectively), with the difference between the strike and the current market price being the profit or loss. Obviously you should not exercise the option if the strike is deep 'out of the money'.