Forward price, delivery price and value of forward contract

Suppose time today is $0$ , and I enter into a contract to buy the underlying asset for $K$ at maturity date $T$ (I am in the long position of the contract). Recall that the payoff at maturity to my long position is $S_{T}-K$ . Here, $K$ is called the delivery price of the contract and must be fixed at the initiation of the contract.

How should $K$ be chosen?
What is the value of my long position in the contract?
What is the difference between the current forward price $F_{0,T}$ and the delivery price $K$ ?

The answers to the above questions are related.

If $K=F_{0,T}$ , then the value of my (long) position in the contract is $0$ . This is true by definition of the forward price; it is the delivery price such that it costs nothing (for both the buyer and the seller) to enter into the contract.
If $K>F_{0,T}$ , then my payoff would be less than that in the 1st case since $S_{T}-K < S_{T}-F_{0,T}$ . The positive difference is exactly $K-F_{0,T}$ . Discounting it back to the present, this amount is exactly $(K-F_{0,T})e^{-rT}$ . I should be compensated by this amount today for the lesser amount at maturity (compared to the 1st case). So I should receive $(K-F_{0,T})e^{-rT}$ (positive cashflows) when entering into the position, so the value of this position is the negative of this amount, which is $(F_{0,T}-K)e^{-rT}$ . This agrees with the formula in the lecture for the value of long forward contract.
If $K<F_{0,T}$ , then my payoff would be more than that in the 1st case since $S_{T}-K > S_{T}-F_{0,T}$ . The positive difference is exactly $F_{0,T}-K$ . Discounting it back to the present, this amount is exactly $(F_{0,T}-K)e^{-rT}$ . I should pay this amount to the seller today for the extra amount at maturity (compared to the 1st case). So I should pay $(F_{0,T}-K)e^{-rT}$ (negative cashflows) when entering into the position, so the value of this position is the positive of this amount, which is $(F_{0,T}-K)e^{-rT}$ . Again, this agrees with the formula in the lecture for the value of long forward contract.